Be careful when you're investigating - there are some Renmei dojo that don't like it if you train at an Association dojo as well. If in doubt, I would suggest going with the BKA over the BKR every time. It's the much larger, internationally recognized organization. From personal experience the Renmei don't believe in things like shiai and small cuts, so your kendo journey may be a little different if you go there. It's possible to be in the BKA and not compete (you have the choice) - but it's impossible to be in the BKR and compete. I swapped from a BKR to BKA club a few years ago. Best decision of my kendo career.
Close but not quite. The tsuka is the handle of the shinai and the tsukagawa is the leather that goes over the tsuka.