1)
Think you played it fine.
2)
On the board is just to draw heavy to check so would bet almost 100% of the time
3)
IP yes 1/3 pot size can be you standard size on flops that hit your range. If the flop is wet however as it is here I just move it up to about 50% as you did. If you have a big range disadvantage (e.g. low to middle cards (e.g. 3c6sTh) and your up against the BB) one can check behind more. OOP you want to check most of your range as a 1/3 pot bet does not accomplish a lot as the IP position player can better realize his equity he can call very wide and even raise some time. OOP a mixed strategy of betting/checking to protect your checking range becomes important.
Always be aware about the tendencies of your opponent if he folds a lot on the flop then ofc raising on almost any flop is good.

This is not a board I would range bet CO vs BU. Most CO 3B calling hands are going to smash this board, and so I would definitely want to be polar here. I would still bet this combo, but I would bet it larger to probably 2/3-3/4 pot.

Probably you are playing vs weak player? Being 200BB deep I don't mind checking flop or betting smaller, there are not too much value going on(your exact combo is rather nice to bet unblocking Ace of spade draw). As played river would be also a bet on almost any card (except 9)

It would be good if you share your own thoughts of the hand first if you had a dinamic with the player or another relevant information.

I believe that being 200 bb deep you have to be cautelous in this flop yea you want to get value from the villian weak range like AJ AQ KT but what happens if he check raise you his range all the ATs KQs KJs T9s even unsuited combos of them and he can semi - bluff you with JTd or JTs so easly if he 3-bets you to 40 bb and you call the pot will have like 100 bb and would you like to play the turn with just a pair and a gutshot (probably the diamond out doesnt count) a 9 T Q A diamond even an 8 turn are bad for you. With this exact combo I'd play check but its fine to bet 25 - 33% but most of the time just value two pairs+ and bluff with JTs FD and BFD and A5d A4d (you can find more bluffs of course) and the size wouldnt be more than 35% of the pot and in the turn you can easily bet big with trips and with some good bluffs blockers like AJ or JT and the reason is that when you bet this small you keep the weaker range of villian in the hand good thing when you got the ATs in your hand or KK.

Just my opinion BTW.

The board is very wet and you have a nuts advantage here. Even 2 pairs would be a bluff catcher for vilain sometimes. I would bet larger (3/4 or pot). And I will bluff my backdoor flush like 98,87. Few bluff since I expect vilain to overcall here. No bluff at all ca be fine I think