"My friend wants to buy that piece of clothing."
If you demand something, 想, 要, and 想要 are all valid. Although 要 would be a bit imposing.
我的朋友想买那件衣服。 Yes, I thought that would be fine too. I’ve reported it just now.
Let me know if you know why it’s wrong (or even just unnatural) to use 想买
Yes it's fine. Some people might prefer 想要买 because the falling tone of 要 "flows" better between the two (otherwise consecutive) 3rd tones of 想 and 买
So the listener knows that the speaker is talking about clothes. 件 can be used for "pieces" of other things like a "piece of luggage" etc. In many cases it would be obvious (in a shop next to a rack of dresses) and you could just say "那件" while pointing and it would be fine, but in other situations you'd need to stipulate 衣服. It's the same in English