Converting my garage into a golf simulator

Help me pick a projector for my garage golf sim
Converting my garage into a golf simulator and could use some advice on projector selection.
My setup:
• Garage: 148” wide × 128” tall × 345” deep (12.3’ × 10.7’ × 28.8’)
• Dual use: Golf sim when I’m playing, park my car when I’m not
• Launch monitor: Foresight GC3
• Computer: Alienware Area-51m R2 running FSX software
• Screen: Carl’s Place custom enclosure (143.4” W × 110.7” H × 59.125” D) – already ordered
• Side barriers: 96” golf room curtains for the walls
• Obstacle: Garage door motor sits about 8 feet back from where the screen will go

What I need help with:
1. Is 4K worth it? My screen is 4:1 ratio (143.4” × 110.7”) – does 4K actually matter for golf sim or just stick with 1080p?
2. Where to mount? Should I put the projector right behind the garage door motor (7 foot throw) or try to offset it left/right to get more distance?
Considering the following short throw (only because the issues described above) (A) AH500ST 4000-Lumen Short Throw 1080P Laser Golf (B) Optima UHZ35ST, am I thinking about these the right way?

3.  Car co-use? Since I’m parking a car in here, any issues I should think about?
4.  Brightness? Garage space with this screen size - how many lumens should I target?
5.  Connection method? HDMI, I want to also hook up a TV, I assume most are using a splitter, and brand / model recommendations?

Budget and other stuff:
Budget’s flexible but want good value. Garage door stays closed when playing so light control is decent. Planning some acoustic treatment too.
Any experiences or recommendations appreciated – especially if you’ve done something similar!