@Nat C.
It seems that you have received great feed back. I have a couple of comments to re-enforce.
I work at a reputable Countertop & Tile showroom in Oregon. I am the Designer and sales person. @Jay Hinrichs We always charge 50% up front - this keeps customers from walking after materials have been ordered and labor performed. However, we are much more lenient with contractors that we have worked with on a regular basis. However, the last 50% is not due till the job is complete and the customer is happy.
That is the worst tile job I have seen in over 10 years in this industry. Yes ALWAYS put tile over backerboard - it is for the stability of the project - never leave it to the next owner to deal with "later down the line" as someone suggested. I replace those tile jobs every month.
For your next contractor hire - ALWAYS make sure they are licensed and bonded ....and CHECK their accuracy - don't just take a lic # from them and assume it is good. Check insurance and bonding always listing yourself as "additional insured". This protects you and should be expected from any reputable contractor.
Good Luck