You can buy them on line. Just Google 'Gateway Arch' and their website will direct you.

We took a Lyft but there was parking on the street

goes all the way to top.

You can't climb to observation deck, you have to ride up make a reservation in advance.

We parked for free at the casino across the river

They only steps you walk up is from the parking on the river.

If you want to see the museum before hand, arrive an hour or two before

I would get there 30 minutes early because you do have to go through security. I also highly recommend using the link in your ticket confirmation email to pre-pay parking. And hour and a half should be long enough u less you want to read each and every sign in the museum area.

Yes. There are metered spaces along the street and several parking decks. We parked in the deck at 4th and Olive, it is $2 per hour and only 2 blocks from the Arch entrance.

Yes. Just call them, there's a possibility of getting tickets. Don't just show up hoping to get them tho- unless you're there right when they open. They usually sell out fast.