How did Hercules travel to all those places in Greece? It must take weeks to go between towns because he goes on foot.
Given that he can out-run a crossbow bolt or a hurled spear and does so many times without being winded throughout the series, and can leap tall mountains in a single bound if his ability to kick people into the air casually is any indication, he's probably got a pretty good clip going on in the overland running department.
The real question is 'How does Iolaus keep up?'
Let The Games Begin: Herc wants to introduce an ethical alternative to war. He sees two boys arguing, "I'm fastest.", "No, I'm fastest.", so they run a race. Herc inaugurates the Olympics saying "You don't have to kill each other to prove who is the greatest Warrior."
Wrong, Herc, wrong. A race proves the best runner. Shooting at targets proves the best archer. But a Warrior's job is War = killing people. It does what it says on the can.
Well, you can do it without the killing bit. You'd still have to fight (wrestling, for example, jousting), but just don't finish'em off.
Fair enough. Fighting with blunt weapons would indeed prove the best warrior, but the games that Herc set up only proved the best athlete. There is merit and virtue in being the best athlete. My whinge is that Hercules said best "warrior".
Wasn't there a boxing match?
You can add all the different contests up and say that whoever gets the most medals has proven, albeit indirectly, that they're the best warrior. If you can run the fastest, jump the highest and the longest, throw a javelin and discus the farthest, outwrestle your opponents, swim the fastest and so on, you've established your warrior credentials. Nowadays, Olypmpic athletes just specialize in one sport instead of trying to win everything, but Herc's working with a much smaller competition and a lot less sports, so the goal of trying to sweep the board and prove you're the best, period, isn't so farfetched.
In "Pride comes before a Fall", the reason given for killing Iolaus is his pride. Later in the episode, Hercules is expressly prevented from helping him fight a HYDRA because if he did, it would hurt his ego. It would be different if the God in question wasn't doing her best to help Hercules because she loves him and in the end helps, but every time she speaks, the core message of what's going on seems to change.
The answer to "Is Hercules immortal or not?" should be very obvious - he's not. The Circle of Fire (third movie) featured the titular fire and its ability to cleanse immortality. Both Hercules and Cheiron spent time in its flames (and the latter is confirmed to have been rendered mortal because of it). As far as this troper is concerned, the big deal made about Hercules's possible (im)mortality from that point on is a Fridge LogicPlot Hole.
I don't think the movies are actually canon though, certainly not any more canon than when we find out that Kevin Sorbo is Hercules pretending to be Kevin Sorbo. Though it's entirely possible that he's like the Norse Gods, ageless but not immortal.
What does Zeus do? Yeah, he is supposed to be king of the gods, but when does he ever fulfill that role. Hercules once or twice criticized Zeus hiding behind the claim of "responsibilities" when Zeus is never evidenced tending to them. Several times, like in the first episode and other instances Zeus is ignoring them and cannot be found by other gods. He only bothers to check in every once in a while and finds out something has occurred and it is too late for him to do anything. Either that or he is shoving his repsonsiblities off on others. The only time he takes an active hand is when he himself is directly threatened. So what does he do? Wander the Earth like it was implied in the films?
A few things come to mind, first we've seen plenty of rain and thunder throughout the series so he does that part of his job. Sure he wanders around a lot but clearly the other gods have little difficulty finding him most of the time and generally respect his authority. It's a major plot point that Ares is doesn't kill Hercules not because he can't but because he fears Zeus. How true that is seems to vary radically depending on when exactly in the series it is. It should also be noted that in Hercules the gods seem to some extent to do their duties passively. There are multiple episodes showing that Ares and Aphrodite are necessary for love and anger to effectively function at all. Death must be free or people cannot die, Prometheus (not technically a god but rather a Titan) has to remain free or humans lose the ability to use fire and medicine. Until Eli's One True God shows up and apparently has the power to keep most of the world functioning sans gods there are lots of hints that suggest that the various Gods need to exist just so the world continues to function.