If you are downloading something that taxes the bandwidth of your internet connection, then it could most certainly affect latency (lag) on other computers or the XBOX. When I download a large file from a really fast pipe (or multiple downloads from separate servers) in the background while playing a game, my pings in the game can jump from as low as 50ms without the downloads to well over 1000ms or more while downloading.
I'm really tired so I'm sorry if my rambling doesn't make sense, but the answer is "Yes, downloading large files can have a dramatic effect on ping times for other users on the network."
edit: To answer the other part of your question, unless the router is physically defective or extremely old and too slow to handle the data you are transferring, the router itself will have no effect on the ping times or latency of individual computers unless you are maxing out the connection with large downloads or other significant network activity. (It has nothing to do with the router, but rather with what you are doing on the router). For example, you have a 6 lane freeway that has the ability to handle a LOT of traffic, but if you overload the freeway with more traffic than it can handle, you get a traffic jam and everything slows down dramatically...