How old is the cat?
How far do you live from her previous home?
Has her paw been 'throughly' checked for splinters, infection, other
injury
or known previous injuries?
How skittish is she around people, and her new home?

That said, despite their sometimes normal crazy behavior,
cats can have psychosis' or other mental problems much like any human
is suseptable to.

A vet visit may be necessary to evaluate her overall condition.
Talk to the previous owner to find out about her previous visits to
the vet first.

As far as letting her outside...
there are 3 schools of thought regarding this.
1. says never under any circumstances
2. says only under certain circumstances
3. doesn't really care where the cat is

I am of the 2nd school, but only because I can be, not because it's
the only right way.

If kitty has plenty room to roam with little chances of being injured,
then it's probably ok.
But a through review of all potential hazards need to be evaluated
first.

i.e. nearby traffic, other non-friendly animals, neighbor hazards,
confined spaces
they could get stuck in, chemicals they could get on their paws, etc.
etc.

This will save you expensive medical bills or regretful burdens later
on.

If you deem it adequately safe, you may want to accompany her outside
the first couple times to reassure her your watching her and that
you're there
for her if she needs you.

If she doesn't groom herself, then you need to start doing it for her.
Buy a brush and/or comb, and brush her daily and talk to her softly
(i.e. she's a pretty girl who has such a nice coat and she should keep
it clean
so she looks pretty...amongst other sweet talk....they love soft sweet
whispers
when you're brushing them)

Other than that, a bath, or sponge bath might be in order.

She needs plenty toys, scratching post, and a little of your time each
day
for play, grooming, petting and 'sweet talk',
to reassure her that she has a home worth staying at.
This may take some time, but it's necessary...it's hard being adopted,
just
imagine you were in her place.
And good luck.

I'd get the cat to a vet.   Something is wrong.  Cats are usually very
clean.