The metronome is an invaluable tool for improving your timing, so make sure you put some time in to this.
If you hate playing with the metronome I strongly recommend backing tracks or drum machines.
You can use a backing track like a kind of elaborate metronome. Rather than using the track to practice your soloing, try some 'in the pocket' playing as an accoompaniment to the backing, not the solo line. Simply put, try beiing the 'rhythm guitarist' when playing to a backing track, not the lead guitarist
Playing in a band will help your timing to an extent, so long as the band already has good timing. There are a lot of variables here. Band playing opens up a whole world of problems usually for the first timer.
I remember having trouble with timing, feel etc when I first played in a group - I also remember never using a metronome back then too so that tells you something!
I can't think of any players who never use a metronome - who knows the answer to this question other than the player himself..?
Another great tool is the radio/tv etc - just stick it on and play along to whatever music comes up - lots of different styles to experience so you need to quickly adjust your 'feel' to fit the genre, and the timing bit is all down to you being strict with yourself.
Sure, learning the drums will help with rhythm, but in my opinion learning the drums will only do that - make you better at the drums. You need to specifically address the problems you think you have with your rhythm, timing and feel on THE GUITAR. The idea of learning one instrument to help with the other is IMO an extremely long-winded way round an issue that can be addressed fairly simply.
Get the metronome out, get the radio out etc and play.