In Omaha hi/lo, your low hand is always determined by using the best combination of exactly two of your hole cards and three of the community cards. link. Suits do not matter for the low and your final hand is the five lowest un-paired cards you can make (again, using two hole cards and three community). The best low hand (or lowest hand) is based on the highest card of your 5-card low hand, where "high" is bad. In other words, 5 low cards that are no higher than 7 is better than 5 low cards where one of them is an 8. Both of these are better than 5 low cards where one is a jack--even if your other cards are super low like A234, the 5th card matters most. In the case of a tied high card, the tie-breaker is the next highest card and so forth until the tie is broken. A low can be split if two people have the same 5-card low hand.
For your specific cases:
1) A5 hole cards + 234 community cards = A2345 low
2) A5 hole cards + 234 community cards = A2345 low (note that when there's any pair, you must exclude one of the paired cards--it doesn't matter which)
3) A5 can be combined with 234 to make A2345. The best that A6 can do is use the 234 to make A2346. By the rules, A5 then has a better low since it has the lowest "high" card in it's 5-card low.
4) Ditto the previous hand. A Q replaces a 2 on the community board, but since there was a pair of 2's in the previous hand, one of them was unused anyway.
5) A2 + 345 = A2345, but A6 can only make A2346 --> again, this result is just like the last two.
6) A2 + 345 = A2345, and A6 once again makes A2346 and loses.
A general rule of thumb that applies to all of these examples is that you can not have a low that's better than the highest of the two hole cards you are using. In other words, with an A6, you can never have a low of A2345 since you always have to use the 6 in your hand. You can definitely do worse, though. Another tip is that when the community board pairs one of your hole cards, you will not be able to count that community card in your low--since low hands are unpaired cards only and you already hold that particular rank, it's a pointless community card as far as the low goes.
This article might be a good read for you.