First time going to an Astros game on June 3rd, what to expect with parking and the Honda Club Level?

This will be my first time going to an Astros game. We are going June 3rd. I've been reading the FAQ and other information I can find online but I still have questions.

Can we watch the game and sit down in the Honda Club Level if we get there early enough to get access?

I am planning to get there at least 30 minutes before gates open. Should I get there earlier if I want access to the Diamond Lot or the others? I don't mind parking at the parking meters on the street if they are close enough to the stadium. I have a 6 month old infant, 2 year old and 7 year old going with me so I'm trying to minimize my parking distance.

With the Sugar Land Space Cowboys, I get free parking with my disabled veterans plate. Does Minute Maid Park offer that option?

If there are any other tips it would be greatly appreciated.