How do neutrons escape nuclei? How do neutrons ejected from a nucleus gain kinetic energy if the they don't repel electrically and the nuclear force only attracts? Does it have something to do with the weak force?
 A: Neutrons are only "ejected" if the nucleus is excited. 
One way to eject a neutron is by simply bumping them off. You could bombard nuclei with energetic particles that could transfer enough momentum to an individual neutron to dislodge it. So the "energy" used here comes from the momentum transfer. 
However, there's also "spontaneous" neutron emission. This happens when a nucleus is very energetically excited and thus, wants to relax into its ground state. This mechanism is called neutron evaporation. A nucleus could be left excited by a high energy collision, for example. When the neutrons evaporate in this situation, their kinetic energy comes from the change of nucleon orbital configuration, where the system rearranges itself to reach a lower level. Nucleons are fermions and they have an "orbital" structure analogous to electrons in an atom. Much of this energy comes from the strong interaction, which governs neutrons and protons and is "blind" to charge.  
Another way to think of it is by using the famous example of $\alpha$ particles being released through $\alpha$ decay. The  $\alpha$ particle is caught in a strong force well caused by the nucleus but if the $\alpha$ particle  gains sufficient kinetic energy through excitations it can quantum tunnel through the strong force potential wall. Here, the strong force is attractive, yet the $\alpha$ particle still manages to tunnel through. Here, the $\alpha$ particle could be a neutron that tunnels through the strong force potential wall. 
Furthermore the strong force isn't only "attractive", there are repulsive components to the strong force as well. It's hard to trivially describe it because QCD is a very complicated problem in general. 
