If someone paid 500K of a 600K loan and then forecloses, what do they get back? What if they paid 100K towards a 600K loan?
Do they get fined for foreclosing? Can they continue the same loan with how much they put towards it if they re-buy the same house they foreclosed on?
Does the .gov help pay for like an apartment or something if foreclosing makes a family homeless?

I think I understand your questions.
When a property is foreclosed, if the amount of the unpaid balance and any fees are lower then the amount it sells for the former is supposed to get the difference. If the amount it sells for is less then the amount owed, there may be a deficiency judgement. Deficiencies vary from state to state.
If you repurchase the property, the old loan will not continue. That loan died with the foreclosure.
Does the government help pay for a apartment? Not that I know of other then section 8 housing and similar programs.
Based on the tone and structure of your questions, I would suggest you contact legal aid or an attorney to discuss your options and to insure you fully understand the process.

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"If someone paid 500K of a 600K loan and then forecloses, what do they get back?"

You need to clarify your post.

If you are talking about buying a mortgage note at a discount and then foreclosing on the property it would work something like this.

800,000 mortgage and you buy cash from the bank at say 500,000.You approach the borrower to work out a new plan to get them on track.They don't want to comply so you foreclose.

You put bid at auction of 600,000.The price gets bid up to 660,000.

You buying the mortgage at a discount just made 160,000 in profit.Of course there are attorneys fees and time line to foreclose and a host of other things that affect this.

The bank has to want to sell the note at a discount to begin with while in pre-foreclosure.

no legal advice