Learn English – I’d like to know the difference between “taste” and ” style”

I think in some ways you have answered your own question(s).

1. Yes, "go shopping". One would not say "go storing". Shopping is, as far as I know, used the same way in UK and US English, and has nothing to do with "shop" vs "store".

2. and 3. You would either name the store or type of store ("I'm going to the hardware store") or, as you say, ""I'm going shopping" or "I'm going downtown", but not "I'm going to the stores". You might be inclined to add specifics: "I'm going shopping for a hammer", for example.

In certain places, you may find "shop" and "store" are interchanged (I certainly do this, having lived in both US-English and UK-English countries). If you want to sound natural, though, stick with what is usual for your area.

Of interest:

eytmonline.com says that "shop" meaning "booth or shed for trade or work" is c. 1300, while "to visit shops for the purpose of examining or purchasing goods" is from 1764 (no idea what they would have said before that!).

Meaning "building or room set aside for sale of merchandise" is from mid-14th century, while "store" for "place where goods are kept for sale" is first recorded 1721, in American English.

There was apparently a strong distinction between "shop" and "store" in US English, with "shop" having retained its original (1300) meaning:

The word store is of larger signification than the word shop. It not only comprehends all that is embraced in the word shop, when that word is used to designate a place in which goods or merchandise are sold, but more, a place of deposit, a store house. In common parlance the two words have a distinct meaning. We speak of shops as places in which mechanics pursue their trades, as a carpenter's shop a blacksmith's shop a shoemaker's shop. While, if we refer to a place where goods and merchandise are bought and sold, whether by wholesale or retail, we speak of it as a store. [C.J. Brickell, opinion in Sparrenberger v. The State of Alabama, December term, 1875]

Not sure whether that use of "shop" remains in US English.

In a formal professional setting, it can be better not to discuss another person's appearance. Some people take compliments better than others and some will read more into it than you mean regardless of what you say.

Complimenting appearance is generally (arguably) more commonly done by women to other women or (less frequently) men. There are times where a male may compliment a female, particularly likely at a special event or out-of-work meeting where their dress is considerably different from their usual appearance. Note that this is a generalization and does not necessarily apply to all people.

If you feel it necessary, I think that the most gender non-specific way of telling someone they look good that won't have any issues with impropriety is to say:

You look nice.

This can seem a bit vague but it can still be very effective if you make it specific to the person you're addressing and how you say it will make it seem sincere.

You can add a bit to it to make it less terse,

You look very nice tonight.

It can also be followed by a specific thing you think they've done well:

You look nice. I really like your dress/suit.

You look nice. That color really suits you.

You could also go this route but be careful to use only if it's really obvious they've made a change to their look:

You look nice. Did you get a new hair cut/glasses?

In general, people like getting affirmation that their new look suits them as they can be a little unsure.

This can all go out the window the second you become friendly with your coworkers/boss and your gender in relation to theirs can greatly affect what's appropriate.