I don't see any reason to donk into the 3-bettor on the flop or turn (let alone check-raise him). You are always getting called by better and very rarely getting called by worse.
Preflop, you are very likely behind after his 3-bet - even with a 30 hand sample, a 30/6 is pretty passive (i.e., he's played 9 hands and only raised twice) and is usually only 3betting with narrow range. Even if you think he is inclined to play back at your steals, these types of players generally do that by calling wider and hoping to hit a flop, rather than 3betting. I would put him on a range of {AQ+,TT+} preflop, though with a high degree of uncertainty. That said, since you are 250bb deep, I think you can call OOP with JJ. If you spike a set, you can get his stack and if the board runs out with low cards, you can often get your jacks to showdown cheaply against this player, who is unlikely to cause you great difficulties if he has AK or AQ.
On the flop, I don't see any point in donking. You don't know anything about villain's cbetting stats, but you can usually expect a cbet in a 3bet pot. When he does cbet, the only hand in the above range that you are beating is AK. However, since he is likely to stab with AK (which makes up about a 1/3 of his range) and you are getting 3:1 to call, I think you can call the flop. More importantly than the odds (in my opinion), is the fact that loose-passive players like a 30/6 tend not to fire 2nd barrels without a hand, so we have a legitimate chance of getting our jacks to showdown. Against a TAG, I think the flop is a check-fold (and you can argue about whether to call the 3-bet preflop). That said, my plan in calling would be to give up if he fires again on the turn.
On the turn, I don't see any benefit to leading out. You aren't going to fold out any better hands and I doubt you are getting any worse hands to call.
On the river, I don't see any point to your lead. Although he showed weakness by checking back on the turn, I don't see any worse hands that call your bet. If he checks back and shows AK, I don't think you are losing any value because I really don't think he is calling with A high on the river. It goes without saying that no better hand is folding to your river bet. If you check, there is also a chance that the villain might take a stab with AK. You can decide whether to call a modest river bet based on pot odds and any feel you have for whether the villain might bluff the river.
When villain shoves, clear fold, as you did. There is a small possibility that he is doing this with AK, but with this villain I think that happens nowhere near often enough to make a call profitable. More likely villain has QQ and TT and checked the turn in the hope that you were on a straight or flush draw that would connect on the river (or that his check would induce you to lead out like you did).