Micky Dolenz ended up playing the drums by default. Davy was too short (!), and had already been pigeonholed as 'the cute one', Peter and Mike had both been established guitar-based singers, so that left Micky! He was sat behing a drum kit that had been set up for a left handed player, given a few minutes 'training' and left to his own devises from then on! so in some of the music video-type bits, he looks completely wrong with the drums.
However, I have been to see them in concert quite a few times (1987 I followed them around UK a bit, then again when Mike joined them a few years later) and he's since learned to play quite well!
The quote that Micky loves to trot out at every opportunity is that the Monkees were maybe conceived as a manufactured thing, but later became a credible group, so he puts it in 60's context: It's as though Derek Nimmoy becomes a Vulcan!
I love the programmes, Head is really strange and quite sad really, as their 'puppet-masters' (Bob Rafelson and Bert Schneider) seem to delight in getting them to say how awful the Monkees were. It's a conscious swan song.
Peter still plays in a band, Davy does a lot of solo shows, and used to be a jockey (he had been training to be a jockey when he was 'discovered' and sent to London to be the Artful Dodger, so he was a 'child star'). Mike seems to be ... well, hmm, err, interesting these days, although he claims to have invented MTV in the past, and he did some really funny comedy and video programmes back in the early 80's (Television Parts and Elephant Parts - check out Neighborhood Nucleur Superiority!!!) Don't really know what Micky's up to but Beatle fans should recall his incidental involvement in the Lost Weekend. He was a child star too, Corky in Circus Boy. (youtube should oblidge).