Do you recomend to purchase airport transfers before you depart or can you just pick them up when you arrive?
For someone going to the Royal Mayan, do you need a car at all or can Thomas More cover most travel requests?
Are there cabs?
Definitely arrange it through royalresorts.com before you depart. It's easy, and less expensive than other services. I contacted Thomas More before our recent trip to the Royals, and they said I couldn't pay them down there but had to pre-arrange it. As you leave the airport, there is a kiosk for transportation - Grey Line, which may be a monopoly - their van service costs more than Thomas More and they also tried to talk us into a pre-paid round-trip taxi which would have cost far more than usual taxi rates. For the way back, we just get a taxi at the front lobby, and usually spend around $20 total for the four of us - cheaper than the van.
Opinions on needing cars vary. We do not need or rent a car, as we spend most of our time enjoying the Tri-Royals. We also don't like the crazy drivers and the number of accidents we see are not appealing.
We use the city busses (cheap and frequent) for an occasional trip into town, to WalMart, to one of the malls, or to wet and wild. We use the free resort shuttle to occasionally go to the Sands. We use the resort shuttle to go to the Haciendas (not normally free to go to the Haciendas, but ask your "private concierge" - AKA salesperson - to send you down and you'll go at no charge, get lunch, and can swim at their beach all day).
I think Thomas More can cover all travel requests. When we go on tours, we have used Thomas More's tour services for a big bus, a van, and once to arrange a private van for a day so we could go wherever we wanted on our own schedule. The private van worked out great - it was around $250/day for the van and driver for ~8-10 hours - went to Tulum, Akumal, the Haciendas, and Playa del Carmen.