Cool thread, thanks spoon.

I wish I could say I didn't have to read to digest to answer some of these questions, but the truth is I did. Also, I haven't looked at any of the responses. This is what I came up with:

1.
We're betting 1 unit into a total pot of P+1, so our opponent needs to fold 1/P+1 percent of the time.

2.
Below is my original answer in quotes
"In order for our value bet to be +EV, he has to call with more hands that are worse than our hand than are better. He's calling with a 7, 8, 9, T, J, Q, K, and A. This means that we can only value bet with a J or better.

When we value bet with a J there are 4 worse hands that call us (T, 9, 8, 7) and 3 better hands that call us (A, K, Q).

When we value bet with a T there are 3 worse hands that call us (9, 8, 7) and 4 better hands that call us (A, K, Q, J)

Therefore, we can only value bet J+ on the river vs a calling range of 7+."

***This is wrong though. The above is true in order for our value bet to be more +EV than checking. I believe that in order to answer the question we need to know the value of P. My reasoning is that we can still make a +EV value bet and get called by more worse hands than better hands because the money in the pot compensates this fact. But, how much it compensates this depends on the size of the pot (i.e. the bigger the pot when he folds the bigger the ratio of worse hands/better hands that call us can be)

3.
Using our formula from #1 where P= ante * 2 = 0.5,

In order to make a +EV bluff we need villain to fold 1/(0.5 + 1) = 0.66 = 66% of the time. So, he always has a range of 12 hands (since there are a total of 13 cards and we have 1 of them). We need him to fold 8 of those hands to bluff successfully, so if he calls with 5 hands then we can't make a +EV bluff.

4.
Not sure if we're supposed to assume that the antes are 0.25 again or not. I'll do it assuming we don't know the ante I guess.

Betting:
When he calls with 8+ and we have the T, he's calling with 6 hands (8, 9, J, Q, K, A) and folding with 6 hands (7, 6, 5, 4, 3, 2).

So, I believe the EV calculation goes like this:

EV = (6/12)*(P) + (6/12) * [(2/6)*(P+1) - (4/6)*1]
EV = 0.5P + 0.1667P + 0.1667 - 0.333
EV = 0.667P - 0.1667 (so if the ante is 0.25 then EV of betting is 0.1667)

Checking:
When we check,

EV = (8/12)*P = 0.667P (so if the ante is 0.25, then EV of checking is 0.333)

I think I messed up somewhere in this solution (to #4) because I did a check on it and it didn't come out right. I've already spent a bunch of time and thought on this so I'll come back and do the other problems later.