So far as I know, you can use quartz as a backsplash but, if you have a high power, pro style cooktop you need a backguard to meet code and to protect the quartz from scorching. I doubt that would be a problem with an ordinary range or cooktop.

Engineered stone is always getting the blame for designer errors. You don't just draw cabinets on paper, you have to coordinate the appliance manufacturer's installation specifications into your design. Too few do this. Show me scorched estone and I'll show you designer/homeowner/installer error every single time.

As far as quartzite staining, it depends on the quartzite and whether the material was labeled correctly as quartzite. Some of the whiter quartzites are mislabeled and should be called marble. Sometimes they call them a soft-quartzite.

I have Taj Mahal quartzite and it is pretty bulletproof. No sealing needed and showed no etching or staining when tested with the usual "bad" products in the kitchen.

You should look up the Geology threads by karin_mt. It will give you a background on the stones, what to look out for, and how to do some tests at the slab yard to determine if you are getting a quartzite or marble. If you can get a small sample to take home, karin_mt has more tests you can do at home to really figure out your stone.

I tested several samples of 'quartzite' that turned out not to be. Besides staining, they actually etched like marble. Luckily I found one that has not had any issues. I like natural stone, so persevered in finding one that worked. True quartzite is very hard and takes longer to cut so you will find that fabrication costs will be higher.

http://usenaturalstone.org/definitive-guide-quartzite/

I have White Macaubus Quartzite and love it BUT I have had some water mark staining... as have others (a search here of quartzite stain - https://www.houzz.com/discussions/query/quartzite-stain will shed light on all the horror stories, a few with a happy ending). There are, however, lots of people who have no problem with their quartzite slabs at all. I don't have any water or other liquids soaking into the top of my stone, like some of those threads, just some small staining around the sink which I think is from silicone, and a little bit on the edge of the prep area and that might be from cleaning with dishwashing soap... I need to do the acetone treatment and then sealer but have been researching my options before I move forward. All that to say, it is beautiful counter top and can make a beautiful backsplash too but it is obviously expensive to bring up the walls and thick (wasn't an option for me b/c then my stovetop wouldn't fit). Plus if the contractor doesn't use the right kind of adhesive, it will bleed through the stone and all the sudden your beautiful (and expensive) backsplash, isn't so beautiful anymore.

Now Quartz as a backsplash behind a stove or range sounds like a bad idea - for the same reasons Joseph mentioned.