Welcome!!
My pups didn't go to the groomer for about six months..I just gave them baths at home. But, hey, I was a little over-protective. I remember reading somewhere that getting their haircut too soon would cause them to turn silver faster, but that was a long time ago and could have been incorrect.
As far as separation anxiety goes, my Maggie was my only dog for the first two years of her life. She would shake and cry when I left and I'd arrive back home to her food not touched and vomit because her belly was empty
This isn't the usual for most doggies, so I wouldn't be overly concerned. I CAN tell you that as soon as I got Max, she never did it again. Companionship helps a lot. A lot of people on here leave the tv or radio on to keep their pups company, I'd recommend trying that!
Again, welcome! You did the right thing by coming here. There is a vast array of knowledge here and it will help you become the best yorkie mommy yet!