Or do they leverage debt, and use the subsidies to cover the equity needed?

Look into a DSCR, Debt Service Coverage Ratio, loan where the rental income can be used to qualify the cash out refinance instead of your personal income.To Your Success! 

If the property can service the debt at 90% leverage between a senior 1st and the seller second, you're looking at a good investment!

From an internet search"Unrelated Debt-Financed Income (UDFI) is a tax that applies to an IRA when it uses debt"

I used $80k to pay off personal debts and $50k to remodel it by myself with my two hands.

(This option would would not carry any debt, all debt would be paid in cash)Or keep our current house, rent it out, and cash flow NET monthly about $1,500?

If a property goes into foreclosure on a second, as Ken said you will either be buying it subject to the first mortgage - meaning it's your mortgage now - or the first has to be satisfied to take possession of the property - or the owner of the first is going to be there bidding what they are owed, so you would need to outbid them to get the property and then you are the person paying off the first mortgage. 

I frequently see where one party is awarded the property and debt and is mandated to sell or refinance it within a certain timeframe to have the other party removed from the note/mortgage.

Now, if the tenant does not vacate after 5 days, then you must go back to the court to get a Writ of Possession.

Since you’re assuming this debt as part of the purchase, understanding the exact amount is critical.