propositional natural deduction proofs

I am not familiar with all your symbols, but generally speaking a proof would be to show that assuming Q is FALSE makes the premises FALSE.
(Method of "backward fell swoop"}

If Q=0 then T =1 & P=1 (THIRD STATEMENT)
thus R =0 (SECOND STATEMENT)
thus FIRST STATEMENT =0

QED

1 (~ T v Q) v R
2 R -> ~P
3 proove:(T & P) -> Q ->I
4 T & P Ass
5 proove: Q ~E
6 ~Q Ass
7 proove: ~R ~I
8 R Ass
9 ~P ->E 2,8
10 P &E 4 contr!

11 ~ T v Q vE 1,7
12 ~ T vE 11,6
13 T &E 4 contr!

I'm not going to post a full proof, mainly because I can't be bothered, but also, I only know Gentzen-like proof systems, which I can't really post on a forum.

Start by using the disjunction elimination on ((~ T v Q) v R). So, now you need to prove (T & P) -> Q from both R and (~ T v Q).

To prove it from R, easy. From, R and the premise R -> ~P, get ~P. Now, assume (T & P) and get P. From the contradiction get Q, and use the conditional intro to get (T & P) -> Q and discharge your assumption.

To prove it from (~ T v Q) you're going to have to use the disjunction elimination again, so you will have to prove (T & P) -> Q from both ~T and Q. You should be able to prove (T & P) -> Q from Q without my help. To prove it from ~T, assume (T & P) from which you can get T. Use the contradiction to get Q, and then use the conditional into to get (T & P) -> Q and discharge your assumption.

You should be able to use the two uses of disjunction elim to discharge all of the other assumptions (of ~ T v Q, R, ~T, and Q).

Edit

Generally speaking a proof would be to show that by using the truth values which makes the conclusion FALSE must make the premises FALSE.
(Method of "backward fell swoop")

CONCLUSION is FALSE iff Q=0 and T =1 and P=1
thus R =0 (SECOND PREMISE)
thus FIRST PREMISE =0

QED (and fairly elegantly !)