Let my start by saying it depends, every unit is different. We all have different expectations for our part time guys and those expectations likely change due to manning, deployments, natural disasters, sickness, health, blah, blah, blah. There are other options besides AGR for full time employment including Technicians and ADOS (if the squadron sits alert).
All these answers are for my F-15C squadron, so take that for what it's worth.
How many times a month will you fly? You need to make RAP (Ready Aircrew Program) - for our inexperienced pilots that is 8 sorties per month. Sometimes you can get two sorties in one day with a tanker or by double turning, but that's not the best training and can't count on that necessarily. We also ask our DSG's to sit a minimum of one 24 hour alert tour each month. When we are deployed or gone on a two week trip we ask our DSG's to either go with us or sit more alert at home since that mission never stops.
All commuting is on your dime.
I wouldn't say we let you pick and choose which days you work per say but we do work together to get the types of sorties that our DSG's need and try to get them in town when there's more flying so they get more for their trip as well as work around other life events and jobs. It's mutually beneficial to work together on the schedule.
If you are an out of town commuter the unit will pay for a hotel for your drill weekends (or makeup) as well as AT (annual training) days. All AFTPs are on your own. If you are on an alert day you'll be sleeping at the alert facility so no hotel then.
You only get paid for days you are on "a status" aka on orders. Your travel day is not included in that other than in some circumstances - but it's few and far between.
Each airline handles things a bit differently as well. I'm a Delta guy so my answers are from that point of view (also I've been on mil leave since the fall of '15 so I don't have the most current info). Anyway, it behooves the pilot and the unit to not totally piss in the Cheerios of every airline out there because we don't want you to be the last person from our unit to get hired at Delta/FedEx/United, wherever.
Most of the major airlines only count your mil leave against your 5 year USERRA limit if your orders are more than 30 days. In theory you could drop mil leave on 7 or 8 days per month and only work the remainder but in actuality that would be hard to swing. Particularly as a young guy (you are asking about right after UPT right?) you'll be working a lot of weekends at the airline. Other than drill weekend (or alert) most ANG guys aren't working weekends. You'll also be working a lot of holidays - federal employees don't work those (other than alert). When I flew the line you could drop one day in the middle of a 3 or 4 day trip and it would drop the whole trip. Not always the case anymore; I've heard of guys dropping a day of mil leave in the middle of a trip and the airline expecting them to be there to fly the other days of the trip. Not a place you want to find yourself. Also airlines are busiest when you want to be home with your family (Holidays, etc) if you drop mil leave over a long holiday (Christmas, Superbowl Sunday, Labor Day, July 4th, etc) they are going to call your commander and make sure you are actually working those days in the ANG. As a commander, I'm not going to put you on orders unless you are actually doing work for the squadron so those days likely won't line up as much either. Like I said, in theory it's possible but not likely that it'll work that way.
If you drop any mil leave some chief pilots will be pissed. There's no magic number.
Hope this helps.