No, I would actually say to keep them with you as you wouldn't want anyone to see you putting them in the trunk or they could easily break in. It hasn't happened to me in the Woodlands but I'm sure it could!
No
No, but put your bags in the trunk or out of site in your car to be safe. Woodlands area is fairly safe, but 5 to 10 minutes south of there is Spring where parked car break ins are a common occurrence.
There aren't public lockers or anything like that, no.
Your car
Thanks! Your answer is awaiting moderation.