[Roland]

I’ve been playing a lot of 2/4 lately, and here’s the way I look at these situations:

4th street:
I would double bet here. The reason is that, in my mind, this bet is a value bet and thus I want it to be as big as possible.
It’s a value bet because 95% of this bunch is way to loose. Most of them will call with anything here, and your aces make money against this anything-range.
Note that you aren’t betting the maximum to try and make him fold. That would be dumb - in my experience nobody folds here, having called the 3rd street completion. Also, you’re not betting for information. You just can’t get information from people who call with anything. (The exception is when you get check-raised and are forced to fold. Then your bet did gain information, quasi as a side-effect).
Lastly, note that the value bet is in order pretty much regardless of your opponent. All kinds of opponents check here - the calling stations, obviously, but also loose-aggressive types. Granted, a loose-aggressive player might be looking to check-raise you on 4th or 5th, but more often than not he’ll have some total garbage hand. And if he does check-raise you, that’s okay, too. You can usually just fold.
(Also note that opponents who bet here are a totally different matter. They are usually the better players who take advantage of fold equity and bet pretty much every hand they are willing to play, usually forcing you to fold).

5th street:
I’d bet again, mainly for three reasons.
First, to win the pot right there. This is almost never possible on 4th, but people do sometimes fold on 5th if they still have nothing going. Obviously, getting them to fold is great.
Second, for value. Although it’s a little less likely that you’re still ahead (your opponent just paid for another card and that card might have helped him) I’d say you’ve still got an edge here.
Third, as a free card play. On 6th street, you might be getting second thoughts about value betting. Specifically, many passive players do the check-call thing with two pairs. So if you think he’s doing that and you didn’t improve you can just check behind. If you checked behind on 5th, he might wake up and bet 6th himself and then you won’t know what to do, like it happened in the hand you posted.
Note that you shouldn’t always check behind on 6th - if you have reason to believe he’s full of it and/or you have some extra outs or maybe your cards are very live, go ahead and bet again.
Value betting and going for the free card all on the same street might seem like a bit of a stretch. But it happens all the time against extremely loose opponents.

Okay, I hope this helps. Note that I had a heads-up situation in mind - in a multi-way pot, things could be quite different.

Returning to your hand - I would call 6th and fold the river unimproved but raise with aces up or better.