Airport transport, accommodation, things to do!

My husband and I are heading to Houston and New Orleans at the end of Feb. We have a few days either side in Houston, which we are really looking forward to.

We have a few questions… – is Uber/Lyft the best option to get into the city? I see there is a bus service, but is this a nightmare when we arrive at 3pm-ish, with luggage, and do you need a bus card for it? – where is the best area to stay that's sort of handy to everything? We were thinking downtown/midtown? – what are the #1 things we should be doing in the 4-5 days we have? (Natural Science Museum and Space Center already on our list) – our flight home is at 9pm. Are there any hotels that let you pay for a 5/6pm checkout or book a half day? Or do we find somewhere to leave our bags and make that our museum day OR just book a room and suck up the full day cost.

Thank you!!