Right now I am in a restaurant with a cycle-rickshaw driver from Varanasi so here are the answers to your questions. Beside this driver, these are my experiences as well as I was there a few weeks ago.
There are only two cremation (burning) ghats in Varanasi, a big one - Manikarnika, and a smaller one - Harishchandra. (A bit of trivia - location of body burning depends on the location of the body, it is carried to a closer one.)
Bodies are being carried through Varanasi streets with body carriers (can't remember what they are called right now) chanting along the way. There are common routes but that also depends on the starting location of the deceased. You can completely avoid this procession by not leaving your hotel at all (since the burning is being preformed 24 hours a day).
I don't really know what you mean by purest ghat but if you mean something like - which has the least dirt or garbage, I would say it depends on the time of the day but if I had to choose one, I'd pick Assi ghat, the southernmost one. This is also an area with pretty expensive and posh hotels (except for a few). If you meant to ask where the water is purest, there is a simple answer to that one - it's the same river so the water is exactly the same everywhere in Varanasi and your parents will be having a holy dip in a holy river wherever they decide to do it. Mind you, riverbed is full of sunken bodies that are put there after the burning is over.
The most busy ghat is Lalita, where there are people praying and bathing in the morning and praying and lighting floating candles with flowers in the evening. Unfortunately for you, it is a ghat that is very near to the main burning ghat.