Q1 - Yes, she should have all her shots before you take her to get groomed (until then you can do the higienic shave at home...)
Q2 - I would say start growing the coat long now..you will need to make the hygienic shave, also trim the ears and paws.
Q3 - The hair around that area should be short, that should be done with the trim (it is what i call the hygienic shave
)
There are different kinds of coat...but the puppy coat tends to be finer than their final coat, so this can be why it looks greasy...just make sure you are rinsing all shampoo and conditioner out of the coat (or maybe even skip the conditioner...)
I hope this helps..