If you simultaneously try to solve the equation of the parabola and chord of contact equation, (note that a=1) then you'll end up with a quadratic in x. The roots of this quadratics are the x coordinate of M and Q. So to find the midpoint of it, youll do (x1+x2)/2= sum of roots/2. Now the sum of roots is -b/a (from ur quadratic u got earlier). So divide this by 2 to get the x coordinate of T. Sub it back into the chord of contact to find the y value.
Since P lies on the line x-y-1=0, y_0=x_0-1 and x=x_0. Sub it into the y coordinate of T and you should get y=1/2 x_0^2 -(x_0-1)=1/2 x^2 - x +1
Soz if it's not really clear or messy, typing this on phone, but ye hope that helps
Edit: lmao just realised I did q2. But ye same thing, to find the coordinates of the midpoint, you simultaneously solve the chord of contact and the parabola to get a quadratic in x, then u do sum of roots/2 = (-b/a)/2. (Cause the roots are the x coordinate of the points on the parabola)
Then use the fact that T lies on the directrix to find the locus
