You're right, Andaman side has the best diving, and this is best time of year for it. The locations you've picked out are good, but quite a lot to try to squeeze in, esp when you consider travel time between the islands. Maybe skip Phuket and spend more time elsewhere before heading to your liveaboard...unless you really want to see what Phuket's all about (in which case 1 night would probably do!).
Some fairly decent diving around Koh Lipe although not the best Thailand has on offer. It's a lovely island to chill and do some easy relaxed diving. Many of the sites are only a short longtail boat ride away and are done as 1-tank trips. Pattaya beach is probably aesthetically nicer, esp further west down the beach, and will be more protected from any waves this time of year (although there shouldn't really be anything significant). We stayed at Wapi on Sunrise beach in October 2016 which is next door to Adang Sea divers - can recommend both. Decent snorkelling around the little islands in front of Sunrise beach too (best on Lipe). One thing to be aware of on Koh Lipe is the amount of longtail boats parked up. They were all on Sunrise beach when we were there as it was the calmer side of the island at that time. There are areas to the northern and southern ends of the beach that are clear though. I believe many of the boats get parked up on the Pattaya side in dry season. Also worth noting that you can walk fairly easily, or get a motorcycle taxi, between the beaches.
Koh Lanta would be an excellent choice. The island's
relatively unspoilt by tourism, so far at least, and some of the sites accessed from Koh Lanta are fantastic! ...Hin Daeng/Hin Muang, the Bidas, Koh Haa, Koh Phi Phi etc. We were particularly lucky and saw a whale shark at Koh Haa, then 2 days later saw mantas at Hin Muang and Hin Daeng.
We stayed on Klong Dao beach which is very safe for children and is popular with families and older couples. The vibe in the evening is laid back, with restaurants and some bars all along the beachfront. Each beach on Lanta has a different character so do your research to find what suits you. We were happy diving with Lanta Diver (March 2014) but you're spoilt for choice on Lanta, some very good operators from what I gathered through research and visiting the shops.
The main beaches on Phuket are very different from the islands above. We stayed on Kata Beach which is much less busy than say Patong, but still a little busy for our tastes to be honest. Phuket's where the package tours head to. There's some good diving from Phuket, eg King Cruiser, Shark Point, Koh Doc Mai etc. Day trips leave from Chalong but all ops seem to offer ‘free’ hotel pick up service. It can be a long day out though.
Koh Phi Phi is, unfortunately, mainly over developed and over-crowded now, although the beaches on the east coast are much quieter. We really enjoyed Phi Phi Relax Resort which has a little dive centre (or at least did a couple of years ago).
I find this website useful:
https://www.travelfish.org/country/Thailand
Tezza also offers some good info, if you have be bothered to read through it all!
Tezza's Beaches and Islands
Another location to consider may be Krabi?
I can't give any advice on liveaboards.
I'm sure you'll get a lot more advice from the SB Thailand experts/residents.
Have fun