This is something I think could make my play relatively significantly more profitable because it's a situation I come across with quite often and am not sure what to do so I decided to analyse what would be best to do and because I've little experience compared to you guys your help/feedback/advice is greatly appreciated. The situation is as follows - and I choose to describe it generally because I want to keep it open for different situations, comments with "it depends on whether..." are welcome, so hands could have also been described as ranges:

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Full 9-max 0.02/0.05NL table, no significant reads/stats yet on the other players nor the other way around and stacks are not very deep or small. Hero leads 3BB from early position with QAo, villain in late position calls and rest fold.
Flop comes A J 5 rainbow. [pot: 7.5BB/$0.37]
Hero bets 5BB/$0.25, Villain calls.
Turn is 8. [pot: 17.5BB/$0.87]

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Stopping at this point, I always think it's very difficult to decide what is best because there are several options and all come with different arguments to justify. Also the range is very broad, but let's try to capture that first: 22+, A2s+, K6s+, Q9s+, JTs+, A5o+, K9o+, QTo+ could be, but could be more if villain would be loose though I think it wouldn't be much more often.
Now we're clearly behind when villain has: AK, AJ, A5, J5, AA, KK, JJ, 88, 55; of which J5 is totally not probable, AA/KK is not so probable (he would (re)raise at flop or turn a lot of times).
We're ahead with all other cards of which is around 80% (it's downright impossible to be very precise) but we're gonna have to eliminate the cards which he would fold to out flop bet, which leaves about 56%.

Let's go through the options and work them out.

1. Checking. Because we haven't got a lot of information right now and his action could give us that. But still, the action could very well be misleading (or am I thinking too much on a 5NL player?), even slowplaying the aces. A significant raise wouldn't be a bluff probably, because we showed quite some strength and bluff c-betting isn't very common from our side. A little raise could mean he would want us to call with a better or worse hand, about 50/50 I guess? A check would mean a (back-door) draw - where we would be ahead - or slowplay with probably something better. A marginal situation, because it's a little hard to be sure what to do next.

2. Little bet (~8BB/$0.40). We are ahead with more than 50% and he would be probably calling it bacause he called the flopbet and both our hands probably didn't improve. So him calling would be maybe nice, but what to do with the river? Because we don't want to keep feeding his stack when he's slowplaying, but if we check there's quite a chance that he bets and it's a very hard call although we could be forced to call it; that river situation is one I hate out of position. If he raises our bet we would have to fold, and if he folds than it was an almost optimal play. Not too bad, only when he calls it gets very difficult because we have no idea what he has, he could be intending a bluff from the beginning, trips aces or anything in between. (Also it seems kinda stupid just repeating the scenario of the flop for some reason)

3. Big bet (~14BB/$0.70). Hmmm, more chances he would fold and that would be great. Calling would make it hard for us as said, a raise would be an easy fold. Almost the same outcome as a little raise. The extra fold value is not worth the extra money I think, a little raise would be enough (or maybe a little more), but another advantage is he will bluff us less often because we're showing a lot of strength. (Btw am I too scared of bluffs?)
I think a bet of around 10BB/$0.50 is best for a bet.

4. Check-raise. We would lose about 50% of the time (his winning range of 44% adding some bluff) assuming we will always fold a re-raise and we raise river if he checks: 32BB/$1.60 (8BB already invested + 10 villain bet + 14 we raise).
We would win the other 50%, with a bet of 10BB/$0.50: 19.5BB/$0.97
That doesn't look like +ev... XD

So I have still no clue after working this out, betting is an alright option but feels stupid, though checking is not better and check-raising is never going to be profitable.
Hmmmm...

Hey Pascal, Im not an expert by any means, but I noticed you hadn't been answered by one of the vets yet so I thought I'd have a go. Take this with a pinch of salt, especially if one of the pros corrects my assessment.

You are a lot further ahead of villains range than you illustrated.
VS unknown I would assume a high frequency of 3bet AA, KK, QQ, and AK pre flop.
Villain's ahead on the flop with only AJ, A5, JJ, and 55, (and sometimes some of the hands mentioned above), and you are trying to get value on the flop from TT-77, Ax, KJ, QJ, JT, J9s.
I would c bet 1/4-1/3 of the pot to keep all these hands in villains calling range. If you bet too big, you will start to polarize his range.
On the turn I would c bet again 1/2 pot to try to get value from his Ax and
Paired J w gunshot hands. You mentioned that if you made a bet and he folded 'that would be great'. This is the wrong way to look at this. You only ever want your opponent to fold a better hand than yours, which he would never do in this scenario. The only other reasons for betting are to deny equity (which his range has little of on this flop) or to extract value. Don't be scared that you are going to run into a monster. The idea is that you play this scenario 1000 times and the combined times you are ahead, you extract more value than you lose on the times you are behind.
Check/Folding isn't an option, and check raising would turn your strong hand into a bluff which is unnecessary with such a strong hand.
On the river, I would make another small bet to get value from his Ax range.
If at any point villain raises you, I would probably fold to an unknown player at these stakes. When an unknown hero raises in EP and starts to barrel off on a static A high board, its a pretty lousy spot for them to pick to bluff raise.
Without any reads, it might be smart to take note of the action and move on. If villain is bluff raising in this scenario, he will be bluffing a lot and you will quickly figure that out in future hands and can value bet him relentlessly.

Thank you, I guess you're right. Best thing on average is betting not too much so he will call with worse hands and raise with better. It just feel like I'm always paying for bluffs but indeed they really shouldn't, and we could take advantage of that later on. Maybe it helps to keep your cbet stat not too high.