Snap! I've just booked to go to Cuba again in March as well - but minus a baby! It's a great time of year to go - hot and sunny but with far lower humidity levels than later in the year so a great time to go with anf infant.
The popping ears and grumpy fellow passengers could be anissue wherever youare travelling to. Ignore the latter and be ready with something for your baby to suck on.
Bottled water is readily available and not something that you need to fret about. As I suggested to the OP, the easiest way of dealing with this is to peel off the label from a bottle here and compare it with the info on the bottles in the shops/hotels. '???? gaz' means it's sparkling 'sin gaz' means it's still and if you avoid anything which says 'aqua mineral' on the label you won;t go far wrong.
A travel cot can be handy to use as an improvised play pen by the poolside - you are unlikely to need to use it as a sleeping cot if you make it knwon to the hotel in advance that you want a cot in the room.
The mosquitos are likely to be less of a problem in March and especially during the day - they come out more after dark and where there is a lot of vegetation. A lot of people assume that any bite must have been a mosquito when often the real culprits are the sandflies that will bite you at any time of day or year on some beaches. The best advice is to take the advice of your local GP and/or pharmacist but something you might want to think about is taking some citronella candles - they do a good job of deterring what my Mum refers to as 'the boodies' of any type. I use a little oil of citronella dabbed on my wrists and ankles but would hesitate to recommend this for a young child but as it is sold by pharmacists they could abvise wether it's something that would be safe to put a drop or two on their clothes for example.
I can't add anything to various suggestions already posted re sterilising equipment other than don't expect to buy anything there.
As for a lightweight buggie, if you take one and leave it there the chambermaids will be tussling for it! They are difficult and/or expensive to get hold of over there. Only you can decide whether you'll manage with a baby carrier harness or not and that will probably depend on how much time you plan to be doing things other than lazing around the pool but I would go with a lightweight buggy option. You've then got something that you can put the baby in to snooze of an evening and take them with you to the bar etc. A baby carrier harness is great for when you are out and about or on the move but can turn into a bit of bind if one of you has to always be 'wearing' the baby.