Preflop call is fine especially with the limped utg. I'd normally 3bet most of the time. It depends on how the table defends 3bet. I'm calling all streets with the min bet on the river as well.

Raising on the flop would force Villain to continue with his value and fold out his bluffs and value hands worse than JJ. If you call, it allows Villain to potentially cbet on the turn with his bluff range. It's possible to you could deny equity to AX, but overall I like calling more than raising. Calling also allows UTG to continue with weaker hands that he would have folded if you raised. It also prevents awkward scenarios where MP 3bets flop.

There are better hands to raise flop with such as sets/straights/flush draws w/o showdown value.

I fear the straight draw so I just call. Do we prefer to raise here?

If you think he's got a SD you want to charge it. So if you think he is going to call a raise with a SD you must raise.

Not sure about calling here, but I guess villain may have a lot of small pairs that we beat

This sentence is quite strange, the street before you put villain on a SD but now you assing he a complete different range.
You should assing a range preflop and the chisel down as actions develops. With your thought process, you removed all the flop assumptions