You've asked three questions. I can give you hints on some of them. I'm still learning and not confident enough that I can answer everyone of them correctly. so just Take it as I'm just sharing my thoughts on them.
So, for the second question You can search 'Reduced Relative clause' and I think u will get ur answer. There are tons of materials available on Google on it.
For the third question, generally article 'The' is used for Definite and Specific things. (specific means a particular thing and definite means one you know already about),it doesn't matter it's singular or plural. Article 'A/An' is used for 'unspecific and indifinite' as well as 'specific but indifinite' things. And in the 3rd question. We have to use 'A' because The Girl there is 'Unspecific and indefinite'... Unspecific means 'anyone'...so here we are talking about any girl.
A violin has four strings.(True for any violin) and it's the case of Unspecific and Indifinite thing. (Indifinite because we don't know which one).
A violin was given to my school. (one violin but we don't which one), a case of Specific and indefinite thing.
For the first one, gerund-participle (-ing) and past participle(-ed generally) non-finite are used for many things. They can be used as modifiers.
Gerund-participle modifiers are used for showing ongoing quality or state Ex - the burning house etc. It can also be used to modify a noun that is the source (agent or cause) of the feeling or emotion. Ex- the intersting book, his amusing behaviour, he was entertaining (gerund-participle as subject complement) etc.
Past participle Modifiers are used to show completed states. Ex- the burnt house etc. It can also be used to modify a noun that is the experiencer of the feeling or emotion. Ex- the encouraged man, the entertained family etc.
In your first question... if we say, 'Mark, dressed in black, walked across the room. It means he is already wearing sth black, the action of wearing is completed.
If we say, 'Mark, dressing in black, walked across the room, it's
giving a sense of ongoing action.