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 fued 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 endenture Han further. We can assume that by the time of Empire Jabba has had to go through with his threats.
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.
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 right 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).
Why didn't the rebel ships fly directly to Death Star's power plant instead of flying through that long trench to reach it?