Answer to quiz: We open UTG and call a 3 bet from the BB. Our perceived range should be fairly strong here, consisting a lot of medium to high pocket pairs (88+), strong suited connectors (89s-kqs), slowplayed big pairs (JJ-AA), and sometimes AK or AQ/AJs.
Once we call his flop bet on Q35r, our perceived range is all our made hands that could bluffcatch (all pairs less than Q), any hand that has a pair of Q or better (overpairs, set of QQQ), and occasionally a float, however, because of the nature of the board, namely, two low sort of dangly cards that can't connect to any reasonable calling range we could have, most of our "floats" are more of value calls with hands like AQ/AK, and our only pure floats might be hands like TJs with both backdoor straight draw and backdoor flush draw (a very minor point, as this is such a small part of our range).
When he checks the turn, his value hands can be discounted somewhat, because the majority of our range will check back the turn (not strong enough to value bet), but it can't be discounted completely, because we do have some floats/might bet some hands for thin value/protection, etc.. When we check back the turn, our range leans heavily towards made hands that will often win at showdown but can't value bet the turn (so overpair hands are discounted somewhat, as well as pure floats can also be discounted, however, if a pure float picked up a draw on the turn, it would be more likely to check back the turn so as to not get check raised off its equity and be able to gain more information by the river as well as to rep a wider/thinner value range if checked to on the river. However, like I mentioned, floats are a very small part of our perceived range, and almost all our range is made up of weakish made hands that are likely to be good at showdown but can't bet for value (as weak as 88 or AJ, and as strong as QK).
Once he checks the river, we have to assume he has a hand that can't value bet the river, since he has no reason to expect us to be that river (no draws existed on the flop that could have missed, we likely dont have a hand we will try to value bet thinly, etc.). Most of his hand range by the river is air, although sometimes it will be air that has some showdown value, with a hand like AK.
The problem with us betting is that we can't represent air very easily, and, if we represent a value hand like a Q with a standard bet, he should be folding almost every hand in his range. To get any sort of a call, we have to represent either a thin value bet that might cause him to spazz-out and raise, or a hand of our own that we are turning into a bluff trying to get him off of a hand like JJ or TT. I don't believe an overbet shove is likely to be called at all, because it looks exactly like we slowplayed a set of QQ or have AA and got tricky on the turn, or we had an unlikely hand like 56s or a5s that tripped up on the river. It is simply too unlikely of a line to take with a hand that was going for thin value or was turning itself into a bluff to call that flop, check back turn, and overbet shove the river (the reason it is unlikely is that, if we had a hand like AJ that wanted to turn its marginal showdown value into a bluff on that river, there would be no need to represent the very top of our range with an overbet, because he likely cannot beat ANY hand that will value bet, so it makes more sense to rep a wider, and thus more credible value range, with a normal sized bet). Because of this, I think a very small bet work work best (1/3 of the pot or so), representing a hand like QJ or JJ going for thin value, and possibly allowing him to shove over, representing an overpair played tricky, or not even caring about what he is representing and shoving anyway. For what it's worth, I dont think ANY bet is likely to get action on this river, and I think the vast majority of the time he will just be folding. But, I feel that this is our best chance to make any extra money.