Hey @Brett Riemensnider, sounds like you've got a fun project on your hands! I fully renovated the upstairs unit (~850sf) of an up-down duplex in Northeast Minneapolis last year, and that project covered a lot of the same items which you mentioned doing for your reno plans.
Window AC. Much like your property, my unit didn't have any central air or heating. For the AC, I went with the GE Profile ClearView window AC unit, specifically the 8,300 BTU model which runs off of a standard 110v outlet, and is available on Amazon with Prime shipping. We only have this one window AC unit in the bedroom of my STR unit, but coupled with a ceiling fan in the living room one room over, it does a really good job at keeping the apartment cool on even the warmest of days. This AC unit was super easy to set up and install, and is also wifi-enabled so you can see exactly how much power you/your tenants are using via the GE app, as well as giving you alerts when the filters need to be cleaned (pull-out reusable filters are very convenient and easy to maintain). I also really like that this is a U-shaped window unit which wraps over the window sill, meaning that you get to keep nearly all of your window's view and natural light, as well as it being theoretically harder for someone to break into the apartment by kicking in/pulling out the AC unit (ours is on the second floor, but the safety concerns for first floor units is definitely a consideration you'll want to factor in).
Heating. In case it comes up during your project, I also removed all of the old baseboard heaters during my reno and installed new Cadet in-wall electric heaters in their place in every room. The heaters themselves (available at the big box stores) are great and keep the unit as toasty as you could ever want it to be during the winter, and they're much less bulky compared to traditional baseboard heaters. If you do install any Cadet heaters though, for the love of god please pair them with the manual knob-style thermostats - the Cadet electronic programmable thermostats are maybe the single worst product I've purchased in my renovating life, and were replaced almost immediately after installing them.
Doors and Trim. Any of the big box stores - Home Depot, Lowes, or Menards - are great for both items. For my money, I think Home Depot has a slight advantage in availability of in-stock door sizes/styles (which you can check online to find your exact style/size/swing and see which locations have it in stock), while Menards has a wider range of trim options.
LVP Flooring. I installed Home Depot's LifeProof LVP flooring into a 120-year old rental house in St. Paul - all DIY installed without much hassle over the course of a few days, and I liked the integrated underlayment that the LifeProof LVP flooring offered. That property's floors are far from level, and as Kevin mentioned above, the flatter your subflooring is, the better your LVP will hold up. 4 years on, the moderately wavey floors in the 1st floor of my 120-year old rental look as good as the day I installed them; the LVP on the 2nd floor, however, needed to be fully torn out and replaced with carpet because there was too much variance in the subflooring and it was causing the seams on the LVP to crack and come apart.
Shower Window. If you're going down to the studs in the bathroom already, tearing out the window trim/stool/apron is probably the best bet. If it's a vinyl window, or if you could swap out the existing window for a vinyl window without much hassle (e.g. if it's on the first floor and you have easy access to the exterior), then waterproofing the inside shower area around the window would be much easier. If it's an old wood window that can't be replaced or blocked super easily, you could always go down to the studs from the inside, put a piece of treated plywood (painted black) over the window itself, and screw that into place so that it's flush with the studs. From the outside, it will look like a dark room, and from the inside, you'll have a fresh slate to install your shower surround (just make sure you don't screw through the plywood and shatter the glass behind it though!).
Shower Surround. In case you're looking for input on shower surrounds, I can't say enough good things about the Mansfield NexTile 60x30x74 Alcove Panel Shower Surround - I ordered mine from Lowes and got it for under $600 with shipping. Installation was a breeze, the panels are super robust and easy to clean, and having an alcove/niche in the shower makes it feel like a much higher end experience for the user than the product's price tag suggests.
My only other piece of advice would be to go keyless with at least one set of door locks at the property. During the reno process, you can send any contractors or other helpers a keypad code to access the property without you needing to physically be there. After the renovation is complete and you're renting the unit, not having to worry about tracking physical keys is a super convenient perk for residents/guests, and also has the added benefit of minimizing the amount of lockouts and other headaches you'll potentially have to deal with later on down the road. Schlage Encode is my go-to for all of my properties.
Hopefully this all helps take some of the anxiety out of the reno process - take it one manageable step at a time, try to have fun along the way, and you'll be finished up in no time!
