Let's break this down into three parts:
1. Should you sell your home to an unrelated business that manages rental property so you can cash out your equity and rent back while you look to buy a new home?
That sounds like it could be a good idea, provided the terms of the lease were generous enough as far as giving notice and so on. I don't think a new lender would look differently at you whether you were renting or paying a mortgage, and while having the cash in the bank for a down payment doesn't change your lender qualifications, it makes your offer more attractive to sellers in a tight market.
2. What if I sell the home to a rental business that I am part-owner of, that will keep it as a rental for other tenants after I move out?
Here, because you are a part-owner, the IRS would be concerned if you were doing things like showing losses for tax purposes, or transferring assets to the company at more or less than market value, or other kinds of improper self-dealing. I would get an appraisal, and make sure all the other partners are ok with this, and make sure the transfer is done at a fair market price. (Like, when Elon Musk used Tesla investor funds to buy his cousin's failed solar panel company at above fair market value -- don't do that!) You will, of course, pay income tax on your share of rental income, and you will pay capital gains tax whenever the LLC sells the house, so you don't want to do this for a long time. (In other words, some of the rent you pay will come back to you as taxable income so you pay twice.)
3. What if I sell the home to an LLC that I alone own, just to temporarily get it out of my name and show cash for a downpayment, and sell it after I move out.
I don't think this is what you meant, but if it is, that's probably a bad idea in lots of ways and should be avoided unless you have super attorneys and accountants to help you stay out of trouble.
