Rental Hunting in Houston near the medical Center. When to start looking, places to avoid, and travel times?

Hi My wife matched in a medical training program in the medical district, work starts July 1st 2023 and we're having a kid in April 2023. Looking to move June 2023 maybe. I know the medical district is near universities so housing is a bit more expensive and living conditions vary from place to place like more university towns. We're moving from the Chicagoland area and have 2 cats that love their litter box in the attached garage away from all noises and distractions. Looking for atleast 3 bedroom 2 bath places.

1. Is coming to look for houses in Jan or Feb too early for a June move in? What time do you recommend? Trying to come early cause travelling and moving with a new born is gonna be hard.

2. I've been using Trulia/Zillow and Craigslist to get a feel and the pickings are small. What website do people usually use?

3. If my wife wants to be a 20min drive from the Medical Center at 6am or 9pm traffic how far out should we be looking? Is Southpark safe and close enough or Bellaire? Google maps is telling me I'd be fine as south as Iowa Colony and north as Mt Houston which I find hard to believe unless Houston has much better traffic than Chicago.