If we stay consistent than we can use left-over materials for the next house.

If you have sufficient equity to pay off the HELOC and have a nice chunk left over then you've increased your purchasing power and only raised your monthly debt obligations a little, or perhaps you even lowered it when you factor in the HELOC payment.

Cash flow positive is nice in that it pays for the mortgage and property management with some left over for future issues, but the upside is in the appreciation on the home and favorable landlord laws in Vegas (relative to Seattle).

That means that there is just no money left over to cover expenses, even if you account for some rent appreciation over time. 

Can there be left-over liens on the property after the foreclosure? 

I guess what I don't really understand is when you refinance do you just get a check from the bank for the total amount (whatever it may be), and whatever is the left over equity basically is the cash I would use to reinvest?

My tuition will be taken care of, but that leaves me with nothing left over for living expenses, so I'm hoping to have continual income from a place I'm familiar with (in this case the city I grew up in) while being otherwise able to focus on my studies.

Therefore, any offer price you make would include the realtors fees with the Owner and likely the leftovers that would have gone to the buyer's agent.  

Because I bought it so low, and added so much value, there was $25,000 left over from the amount the bank would loan on it.  

Why would you pay that loan off completely and start over when you can leverage leftover money you save each month and purchase more real estate?