If you’re patient and can go over budget, an OLED might not be totally out of reach. I snagged a 77” OLED for ~1600 USD earlier this summer (refurb from third party reseller). You might be able to find one close to your budget if you drop down the size a bit.
I think we’re about 8ft? I’ve heard there’s no minimum really. Basically you don’t want to have to physically turn your head to see either side… But beyond that the closer you are the better you see the picture.
Aiui the distance thing was more back in the day where if you sat too close you’d notice pixels. But with modern resolutions that’s not a problem anymore.