Yay for a first trip!!! I love that it’s popped up in less than a day and that as you mention, in a week you’ll be done & home from it. Impromptu DL trips are definitely ones I have experience with & so much whimsical fun!
Quick note on your priorities before jumping into your specific questions, Haunted Mansion will be closed for a refurb on your dates (and might already be closed now, I just did a quick check for next Thur specifically).
Good news, though, on the inside it is very much the same ride as WDW’s with WDW having the superior queue with all the story and theming, there’s much less of that in DL’s outdoor queue and most notable of what’s different about DL’s HM can be appreciated by looking at the exterior facade of the mansion which you hopefully might be able to still see behind the gates.
A great substitution for HM is Pirates as DL’s is most definitely different (longer with an extra drop and extra scenes not included at WDW).
Ok, now to questions!
Short answer, no. Before park opening (and maybe even the first hour or so of park opening?), the monorail is reserved for onsite hotel guests. Because the monorail puts you beyond the checkpoints in the park where they check for an onsite reservation to proceed to the queues open for that early entry, they will check before scanning you for entry to the monorail for an onsite reservation.
All the turnstiles open up at the same time for anyone waiting, onsite hotel guest or not. Those who have early entry will be able to show their reservation at checkpoints that will get them past ropes towards the lands open for early entry. The rest will wait behind ropes usually put up around the hub, but could be at the very end of Main St.
My experience since they’ve moved to a 30 min. early entry is that the turnstiles don’t start letting guests in until it’s 5 (or less) min. before the early entry starts & most times it really is just at the early entry time. But I’ve seen others say the turnstiles will open as much as 10-15 min before the early entry starts (so for an 8am opening sometime between 7:15-7:30).
Because you’re parking, you’ll need to wait for the parking lots to open (7am, or one hour before regular park opening is what they advertise but it can be a few minutes earlier or later if they’re delayed/running behind). It’s not uncommon for a line of cars to start forming at 6:30-6:40am either. Once you’re through the parking part and then walk (or tram or shuttle) to the turnstiles, it will be 7:15am at the earliest, and potentially closer to 7:30am. So I would go right to the gates from the parking to get in as close to 7:30am as possible as that’s when the people will be starting to scan in (and if they have G+ getting those first G+'s LL’s).
There’s a great thread on all things G+ for DL here but I’ve included below the general order of priority for each park listed in there which I agree with (it is missing MMRR but as that’s not one of your priorities, I won’t try to figure out where I think it should go but somewhere around MFSR/Matterhorn area).
If I had to combine the parks and then factoring in your priorities, I would probably say
IJ > Space > Webslingers > GotG > Matterhorn > Incredicoaster.
None of them run out terribly quickly but IJ, Space definitely run out in the middle-ish/afternoon part of the day (GotG if it’s not running all elevators/lifts can run out before Space, so just keep an eye out on if times are moving faster than Space’s and/or if the wait times are brutally high like 90-100+ minutes that usually means they’ve had to cut down on running 2 or more elevators, and if they are, still grab Space 2nd since you’re starting there & grab GotG right after tapping into Space to start the DCA stacking). Webslingers and GotG on a normal capacity day will run out too but usually after IJ and Space. Matterhorn and Incredicoaster can run out but are definitely more available and have decent LL capacity.
Another good reason to get IJ first is if it goes down for your window, it will turn into a Golden Ticket and if you use it right, it won’t matter what time you book or when you show up to anything for the rest of the day. But it is complex and I don’t know if I fully understand it to advise on how best to use it, so if you don’t want to dive into the technical details of a Golden Ticket and G+ in the advance G+ threads, then ignore this bit. But IJ is still a good bet to get first because it does run out the soonest of all the LL’s.
Quick service and snacks are good allllll over so no need to advance plan food. If I had to recommend just one QS in each park it would be Cocina Cocomonga in DCA and Bengal BBQ for a smaller lunch or Plaza Inn for a bigger portion in DL. But there’s so many good ones that that was a hard choice and I would put anywhere you can get a corn dog over Plaza Inn (Little Red Wagon & Stage Door Cafe in DL and Corn Dog Castle in DCA). We also love Award Weiners, Pym’s Kitchen (for lunch only), Shawarma, Aunt Cass’s, Cozy Cone #3, Lucky Red Fortune in DCA and Harbor Galley in DL. We do also take every opportunity that we can to snack too (churros, beignets and any flavored Dole Whip treat from Tropical Hidaway or hand-dipped customizable chocolate bars from Clarabelle’s). And since I’ve now given you about 5 days worth of food choices in DL, good luck narrowing it to one day of food 
And the second thing to know is DL is a lot more “in the moment/go where the wind takes you” so my day always starts out with my plan being more a checklist and as things change we don’t go in order, but follow the flow of how they’re changing (i.e. we didn’t plan to do Incredicoaster until 5pm but right now it’s a 20 min wait and we’re not far away, let’s go there next instead").
I have no experience with LAX (except on a connecting flight to somewhere else) but tagging @Jeff_AZ in case he knows from LAX experience he has. And @Wahoohokie in case she’s ever done a rental and LAX (I’m pretty sure she has).