From Edmonds' notes on Transformation groups: http://www.indiana.edu/~jfdavis/seminar/transformationgroupsb.pdf

Problem 9. Show that if a group $G$ acts on the torus $T^n$ with a fixed point x and induces the trivial action on $\pi_1(T^n, x)$, then the action is trivial.

So the answer to question 1 is yes. Question 2 is answered by @YCor's comment and also discussed in the above notes.

From Edmonds' notes on Transformation groups: http://www.indiana.edu/~jfdavis/seminar/transformationgroupsb.pdf

Problem 9. [p.28] Show that if a [finite] group $G$ acts on the torus $T^n$ with a fixed point x and induces the trivial action on $\pi_1(T^n, x)$, then the action is trivial.

So the answer to question 1 is yes. Question 2 is answered by @YCor's comment and also discussed in the above notes.