Problem 1:
Approach: You calculate Tension forces on blocks by dividing applied force as F->F/2->F/4. Now calculate acceleration of a block is F/4m – g. If it is less than zero, a = 0. Acceleration of a pulley is equal to (a1+a2)/2 where a1, a2 are accelerations (rate of change of change of length) of two sides of string.
Problem 2: Your approach is totally correct. You got the correct constraint equation.
If m1 is moving to right and both m2, m3 moving downwards, you apply a1=(a2+a3)/2.
Also keep in mind that these acceleration are in the reference frame of earth.
Please let us know if that clarifies your query or you need further explanation.
Thanks,
Sandeep