Why did Luke take charge during the final Trench Run? Biggs and Wedge both have more experience than Luke, and they outrank him as well (its unlikely that Luke was given a high rank for his second mission in the Rebellion).
They don't outrank him - they are all regular pilots. And Biggs, at least, knows about Luke's piloting skills. He took charge because he was the best pilot (and because he was ordered to by Red Leader).
As already said, the pilots with lower numbers don't outrank Luke. When Red Squadron was originally formed this would have been true, but casualties leave gaps and mess up everything, Red Leader/Red 1 being the only exception. Rebel squadrons in the EU fly in pairs. Judging by dialogue Biggs (Red 3) was Luke's (Red 5) wing-man. By the time Luke made his run on the exhaust port, most of Red Squadron was wiped out. Wedge's wing-man was presumably shot down already, and Biggs was already watching Luke's back while Luke looked for more TIE fighters to blast apart. Hence, Wedge would take up position on Luke's other wing, allowing the trio to share shields. This particular formation is not really used in real life in combat due to being awkward, but here would be a case of reorganization. The Y-Wings of Gold Squadron would presumably be organized in the three-ship formation from the start since they were supposed to be the ones to blow up the Death Star in the original plan. A good chunk of this is fan speculation, but the rest comes from the EU. I hope this helps.
Plus, sometimes, during a crisis, it's not always the person with the most seniority or authority-by-position who takes charge, but sometimes someone farther down the chain who seizes the initiative. At this point, most of both squadrons have been blown to pieces, so there's little organization left of any kind. Luke stepped up, so he got to be in charge.
Why is Han Solo on Tatooine? When I first saw the film, I assumed he was hiding out on this backwater planet in order to avoid the intergalactic crime lord that he'd pissed off so much he was sending bounty hunters after him. Imagine my surprise, on seeing RotJ, when it turned out that Jabba actually resided on Tatooine. It's not as if Han has gone there to apologise or offer compensation for the cargo he dumped. He's simply whiling away the hours in a dingy cantina. He's not even surprised when Greedo shows up to threaten him, so he obviously knew he was in trouble. It could be likened to someone ripping off the Vegas mob, then coming back to town to play the slots.
Tatooine is a whole planet, and Jabba is just one mob boss not some all-seeing overlord. Han probably expected to go unnoticed. As for what he was doing, he was looking for work - a wretched hive of scum and villany probably has need for smugglers, it's also possible that he had recently taken a shipment to Tatooine.
It's more like someone ripping off a Vegas mob, then hanging out in Reno.
Han was quite capable of making another deal with Jabba in the first movie and the expanded universe suggests that Greedo was more a sign of Jabba's irritation than any real grudgenote until Han disappeared for a while without paying him off.
It also is probably a case of Greedo having an already existing feud with Han. It makes complete sense for Jabba to threaten to kill Han if he doesn't pay his debt. That's how mob bosses keep people in line and paying their debts. But unless he has defaulted many, many times and seriously pissed him off, Jabba is much more likely to want Han brought in alive so that he can take the Falcon as compensation and possibly indenture Han further. We can assume that by the time of Empire Jabba has had to go through with his threats.
No "probably" about it; Greedo out-and-out says he's been waiting a long time for the chance to shoot Han. Jabba almost certainly sent Greebo to intimidate Han, but due to a pre-existing grudge, Greebo took the opportunity to provoke Han and incite a violent confrontation. (One he obviously hadn't expected to lose.)
Based on the extended special editions, minus the fact Greedo shot first (NO!), there is a scene where Jabba is at Han's ship waiting for him. The whole scene leads to some Fridge Brilliance to me. Greedo was set up to die by Jabba. Reasons are not necessary, just the fact that we know (later) that Han is such a Bad Ass and the fact that that scene was so "cordial" leads me to believe that Han might have been there to talk to Jabba specifically... Jabba just surprised him a little by showing up first. He always seemed to have the intent to pay Jabba, things just kind of spiraled out of control and eventually, Jabba just wanted Han dead.
This is explained in The Han Solo Trilogy. Han had come to Tatooine specifically to see Jabba, to ask for more time to pay off his debt. He thought he'd have better luck talking to Jabba in person.
Why didn't the soldiers fire on the pod despite knowing that there could be droids
Presumably they didn't think that if there were droids, they'd be worth shooting at. I doubt the guys at the guns new yet that the Death Star plans had been jettisoned from the blockade runner.
The commanding officer specifically told the gunner to let it go as there were no life forms detected and it had probably short-circuited.
One would think that that would make it even more eligible for target practice...
That'd be like the USS Missouri firing at an empty lifeboat during a combat operation.
If you see an empty lifeboat, you can be fairly confident that there aren't invisible sailors on it. But within the context of star wars, they should have known that droids don't show up on life scans. Which raises the question as to why they just scan for life forms instead of droids(assuming that's possible).
It may not be. What would you scan for? Electronics? In an escape pod that's filled to the brim with them?
Honestly, the more sensible reason they didn't fire on it is because if they blew it up, they would never be able to confirm whether or not the plans were there, and that was the mission—to secure the plans.
Doesn't C-3PO have a line to the effect that droids aren't allowed in escape pods? Presumably R2's programming to find Ben had over-ridden this but the gunners would have no reason to think it.
They do show that they have droid detectors in the Cantina, but aside from that, when your commanding officer tells you not to fire, you obey them. As for why he told them not to fire, who knows. Maybe he has to write a report on each firing incident and didn't want to do the paperwork. Not even being sarcastic, considering that the series seems to have a great deal of bureaucracy involved in every decision made (or not made).
Blowing the pod apart wouldn't necessarily be a smart move, whether there were life forms on board or not. They want to find the stolen plans, after all; if they vaporize the pod, they'll have no way of knowing if the plans were inside or if they were smuggled off Leia's ship in some other way.
Why didn't the rebel ships fly directly to Death Star's power plant instead of flying through that long trench to reach it?
Because flying directly would've exposed them to fire from every turret on that side of the base. The trench provided cover.
Why didn't Darth Vader recognize Leia when they were in the cell together?
I know the real reason is because the script hadn't been written for RotJ yet (or even really conceived) but what's the in-universe explanation? Vader apparently figured out who Luke was pretty quickly, knowing without a doubt that it was his survived biological child; why wouldn't he have done the same for Leia? Or at least noticed while she was resisting the mind probe torture that she was using the Force, if only instinctively to protect her memories. He would have wanted her guarded even more and possibly it would have made him doubt the Emperor's lies about his dead family at an earlier date.
It's instructive to note that Vader only recognizes Luke as his son after Luke Skywalker's name has been made public in-between ANH and TESB as The Hero of Yavin. Vader shows no recognition of Luke on the Death Star anymore than he does Leia, and he did see Luke face-to-face in the hangar bay, and could only pick up 'This one is strong in the Force' as opposed to 'Wait a damn minute, that's my kid!' when sensing Luke's Force presence during the Battle of Yavin. Since Leia is not going around using Anakin's own last name, and is far more "latent" in the Force than Luke is (Luke was already actively using his powers even before he was trained, while flying around on Tatooine — Leia wasn't doing anything similar), the clues that let Vader eventually pick up on Luke aren't going to tip him off about Leia.
Their confrontation on the Blockade Runner obviously isn't the first time they've met. It's likely that Vader's encountered Leia before in her role as an senator for Alderaan, detected a very slight affinity for the Force in her, and dismissed it as just another case where a petty sensitive's unconscious knack for reading emotions got them a leg up in politics. If there were still a Jedi Temple, her talent might've been worth testing, but the Sith don't recruit anyone at the weak end of the spectrum, so why bother?
He probably didn't know Padme was having twins.
So what, exactly, happened to the moisture farm? Luke sells off his landspeeder, but there's no indication he sold the farm too. Did all of the droids and vaporators and other equipment just get abandoned for the next sandcrawler full of Jawas to swipe?
Why is it that everyone in the galaxy appears to know the language of everybody else? Think about it — Luke speaks droid (whatever that is), Han speaks fluent Wookish, Huttese, and whatever language Greedo speaks. Where did all these hardworking moisture farmers, starship pilots, and mercenaries find the time to learn three or four languages? And how come everyone they speak to in their own language understands them without having to use a translator?
Everyone in the galaxy clearly does not know the language of everyone else. Luke speaking a language of droids that he works with and Han speaking languages of the people he works with don't mean everyone is fluent in everything. If they were, droids like C-3PO (whose main function is translation) wouldn't be needed.
Farmers, pilots, etc. knowing several languages isn't all that strange anyway (See immigrants, multilingual countries, and probably other examples on Earth.)