I think so! I also heard, talking to them a lot or once per day, and doing tasks for them and never hitting them with nets or pushing them helps with the relationship too, but I dunno if all of them are true xD
Letters definitely do. I always mention "friend" in the body of the letters i send them. I dunno if mailing gifts does anything, they've never acknowledged any gifts that I mailed but they'll talk about letters I sent them. Also, they'e never worn clothing I mailed them or displayed furniture i mailed them either for that matter, it's like the gifts went into oblivion...
I've always thought that gifts helped improved friendship. In experience every villager that i've spammed with gifts has always become my bestfriend. So yes, I think it does!