I would like to install TOR as relay on a hosted personal server. I have loads of bandwidth that I don't use. It's not an exit point. Can this hurt my server somehow? Possible problems I'm thinking of are blacklisting the IP-address, or something similar.

I know that exit points get blacklisted on many servers. So if I'm using Tor as a client, I will probably use a blacklisted IP-address for the outside world, so cannot access those sites.

However, I'm running this on a server, and as a public relay. Could this hurt the functioning of and access to websites on this server?

I could install it as a bridge. I'm a little confused about the difference between bridging and relaying. If I understand correctly the only difference is that a relay is public. Does this mean that bridging only works if I know someone and give them my IP-address?