They are 2 separate transactions, with separate costs.You could possibly save on some of the small title co. admin costs, but I cant imagine it being more than a couple hundred bucks

Since I’m using conventional financing I was worried about a foreclosure auction since it might not allow enough time for the title & appraisal needed by the bank,and I would be competing with all cash buyers who would win against me.

However, you contact the title or escrow company and they may be able to provide you with one.

Some attorneys and title companies don't like quit claim deeds, so have them prepare the deed they will insure in the future. 

This title company’s settlement fee is $1200.

The issue of course is that you cannot take title to your new property before your old property sells - you can't buy from yourself. 

If that is the case, do title companies and other parties involved who ensure the transaction goes through care about the terms at all?

At a minimum, you need to verify their identity and their criminal record.I generally combine their income to meet my standard of 3x the rent but if there are three or more unrelated people renting together than I require each individual to make a minimum 2x the rent.

We want to buy together, then refinance and take myself off the mortgage and the title.

I selected the title company that they recommended because I figured in this situation it may be easier.I have purchased several homes and typically the settlement fee from the title company is $500 +\- $100, I was expecting a similar cost.