To answer in order: $200 is enough to move up to 10nl if you have beaten 5nl and 10nl before and you're starting over with a new bankroll. If it's your first bankroll and you learn as you go along it's probably wiser to wait until $300.

Pre-flop: Limp and min-raise before you - I would raise this up to $0.35-$0.40, hope for button to call and get headsup to the flop.

Flop: I accept checking. A paired flop is a bluffing flop because there are only two card strengths on the board. If you have an unpaired board 9 cards if in someones pocket holding will make a pair - if the board is paired only 5 cards will make someone a pair (or trips) and J6 doesn't lend itself to any straights and no flushes are possible. I don't hate calling as much as others. Traditionally calling a bet AND a raise screams strength at the top of your lungs making it very hard for your opponents to put more money in - raising is seen as less strong. That said, the total bet size AFTER the raise is less than 2/3 PSB so you can be forgiven for treating it like a single 2/3 PSB. I don't hate calling to keep people in here so you can extract more. However, I also prefer a raise here - probably to $0.70 or so. It seems bluffy while at the same time giving fair odds ($0.50 to win $1.32 for button if MP folds). Given a call here I would plan to lead out with $1 into $1.82 on the turn and $1.51 into $3.82 on the river. Alternative plan I guess would be to check/call any turn bet and open shove river regardless if any turn bet was made.

Turn: Time to play hardball. With the 3 sixes on the board anyone with a high pocket pair or AK/AQ will think they are in good shape and think you're full of shit if you bet. Your full house at this point beats any hand that doesn't have a 6 - and button hasn't played as if he had a 6. But he HAS played as if he likes his hand and wants to extract value. I would probably outright bet here. Button has $3.01 left at this time and to get all-in in two bets I like the divide by four rule of thumb which indicates a turn bet of $0.75. This is a bit big relative to the pot so may see some AK hands folding. I'd modify that down to $0.60 and if any A, K or Q comes on the river I'd open shove $2.41 into $2.02 (slight overbet) as any lesser full house will still call - if no A, K or Q comes on the river I'd probably bet $1.6 into $2.02. Given the check-with-a-chip move (used to give you a chance to BLUFF at the pot - I read this as strength) I would definitely raise to $0.75 at that decision point.

River: As played, fold. If he does not have an A, K or Q calling will let you split the pot - so he needs to have A, K or Q no more than 30% of the time for calling to be profitable. As the hand played I'd say he has A, K or Q at least 80% of the time.