For short distances, CAT5e will work just fine. The primary benefit of CAT6 is that it’s built to higher tolerances, making it less susceptible to crosstalk and other noise. But generally that’s only a concern as the line grows longer. I certainly wouldn't run out and purchase CAT6 if I already had CAT5e, just a waste of money in most cases. The only exception might be if you planned to run ethernet through the walls. In that case, it just makes sense to future-proof the installation given the effort/cost involved.
I'm not sure what "bottlenecks" you might be anticipating. As long as you’re using Gigabit adapters, you should attain Gigabit speeds. If you were *really* concerned with squeezing out every last bit, you might be better off w/ a standalone Gigabit switch. Most routers use only budget-grade Gigabit switches. For example, they likely don’t support jumbo frames, use smaller and fewer buffers, have a relatively small backplane (so concurrent throughout maybe compromised), etc. But like CAT5 e vs. CAT6, I wouldn’t opt to upgrade to the higher-grade solution unless I had specific problems. In most cases, the budget solutions will suffice.