Hrothgar42

We set up a home gym this fall, and have been using it a ton. Here's our setup:
Bar and weights: Vulcan one basic bar, 240lbs alpha bumpers, Hbasic squat stand, Rogue infinity spotter arms, Titan weight tree, American Barbell change plates
Other things: A couple powder coat kettlebells from kettlebell kings, slam ball, balance thing, jump box
Even in the garage, we're using floor protection: Two 15'x4' 8mm rubber flooring rolls, Titan silencer pads

My sister also has a home gym setup, which is entirely Rogue. She loves it. She also made an 8'x8' platform for oly training.

Questions for you:
What types of lifts will you do? Deadlifts, squats, and power cleans are my typical lifts. Power cleans and olympic lifts (snatch, clean and jerk) should be done in front of a rack. (You need 4' for deadlifts. 6-8' for oly lifts). With a squat stand, you can remove the spotter arms and use part of the space in the squat stand footprint as you lift.
To make a rack exceedingly stable, you'd bolt it down. Given that you're in the house, that would be bolted to a platform. (Recessed bolts on the bottom of the platform, probably rubber under the platform to protect your floors.)
Are you going to bench press? If not, you probably don't need a bench.
How much are you planning to lift? I'm comfortable with a squat stand over a rack, but I weigh 150lbs, so I'll never lift a ton.
If you won't do lifts where you drop the weights, you don't need bumpers. You can just use iron plates, which confusingly are often called olympic weights, even though they're not used for olympic lifts...

Tips:
Use masking tape to mark out where you want to put things. It'll take more room than you think. Remember you need space to maneuver weights outside of the edge of the bar.