This is question is originally in context to elasticity (young mod, bulk mod etc). but i want some clarity on the tension forces (or torque)

Simply put, express the total torque on the rod around some point if your choosing; it should be zero. Then express the total vertical force on the rod; it should be zero too.

Say you pick the point where $m$ is attached as reference. $T_1$ adds a torque about this point in one direction. $T_2$ adds a torque in the opposite direction. They need to cancel each other out since it's a static situation. Then there are the linear forces - two up and one down. They also need to cancel out. Now you have two equations and two unknowns - solve away.