In the sentence,
I asked her if she was going shopping and could get me some toothpaste
Is shopping a noun,verb or a gerund?
Can we put two verbs together like this: “I am going swimming this, Friday.”
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In this particular case, "to go shopping" is part of a broader pattern of "to go X-ing". Other examples include "I'm going bowling", "I'm going fishing", "I'm going running", etc. I would classify shopping in this case as a gerund.
To answer your second question, most verbs that can be used in this pattern are leisure activities (hiking, swimming, riding). One interesting case is that "to go drinking" is fairly common, while "to go eating" is not.
See the wiktionary entry for go, definition 30.
Some grammar books don't use the specialised term 'gerund'. Michael Swan, in 'Practical English Usage' (3rd ed, 2005, section 293 and following), calls it the '-ing form'. He writes:
We can use -ing forms (e.g. smoking, walking) not only as verbs, but also like adjectives or nouns. Compare:
You're smoking too much these days. (verb: part of present progressive)
There was a smoking cigarette end in the ashtray. (adjective describing cigarette end)
Smoking is bad for you. (noun: subject of sentence)