Install external antenna to router with only internall antennas?

The easiest solution may be to just buy a router with external antenna connectors.

It depends if you have to unsolder the antenna connections from the main board. If they use the small plug connectors you can buy a short cable and connect it to the external. If you are going to solder have take large chance to damage something.

Problem is these cables are expensive since they are microwave cable and all the connectors you will need are not cheap they are very specialized for the wireless market....mostly to prevent you from doing exactly what you are trying to do.

Will it make a difference, most likely not in a home environment. The walls and floors will just absorb more of the signal. The only way this tends to be effective is if you use direction antenna on both ends so concentrate the power in narrower beam. Still it does not take much when you consider the metal film on windows that you can see though can completely block wireless signals.

I have a Amped SR10000 that I'm using as my client bridge. Would it help if I added high gain direction antennas to that even if the main router (e4200v2) doesn't have a directional antenna?

You can run direction antenna on only a single end. It should help but antenna theory and design is as much a art as it is a science.