I went to Singapore last year for 4 nights and managed to do everything you did and a bit more. I went at the beginning of June, and only had a couple moments of rain. But prepare yourself it is very hot, all year round. Based off google, the months with least amount of rain days are Feb, March and June.
2. I would suggest 6 to 7 days, to be able to do everything you have listed. And just plan to do the major attractions on a weekday. The day trip to Legoland Malaysia is possible too.
3 and 6. The best, cheapest and easiest way to get around is to use their public transport. The MRT network is awesome (trains and buses), and you can get to all attractions using it. You can get a 3 day tourist pass for $20, and that’s unlimited transport. You can get it at the airport and some of the larger stations. So you buy it, and then after 3 days just buy another 2-3 day pass. Apple and Google Maps has the public transport for Singapore built into them, so you can easily figure out how to get everywhere using them. For attraction websites, just google what you want. For anything on Sentosa Island, just google that name and all the info for Universal, aquariums, etc are on there. Note that transport to Sentosa is a seperate ticket though, but you can just walk to the island
4. You’ll be able to get all 3 of those zoos done in one day, but yes definitely a long day. And universal will just be one day too. You probably won’t need fast passes either. But if you were wanting to see the parade, it’s only on certain days.
5. Everything you listed you’ll love. Be sure to check the Gardens by the Bay out at both day and night. And when you go to Sentosa, there is a Crane Dance show on the water at 8pm every night, highly recommend that
6. I would recommend looking at the ibis hotels. I’ve been twice and stayed with them both times. They have 3 hotels, one is right next to the train station, but costs more, the next is a few hundred metres walk to the station, and the 3rd is about a 1km walk, but is the cheapest and has a pool.