RC circuit with current source
I think that to make Kirchoff's voltage law hold I need to consider voltage drop across current source too. But, I'm not sure. Anyhow, are my expressions for VC(t) and VR(t) correct?


Exactly this. There is a voltage across the current source as well. This voltage times current equates to power consumption of the circuit. A correct Kirchoff voltage law goes: $$Vi = Vr + Vc$$
where Vi is the voltage across the current supply.

The solutions for Vr and Vc initially given in the question are almost correct. The integral solution for Vc must take into account the capacitor's initial voltage. If it is zero, then it is as given.