Context, context, context! Either 1 or 3 could be correct.
1 is employed in a context where your discourse concerns the time when she was living in London and mention her move to New York only to provide a 'timeframe' for her London residence; for instance:
She was living in London for several years before she moved to New York, and found British accents (once she was able to understand them) much more attractive than that of her native Minnesota.
3 is employed in a context where your discourse concerns the time when or after she moved to New York and wish to mention her prior residence in London as the source of some state which obtained at that later time; for instance:
She had lived in London for several years before she moved to New York, and found it difficult to understand even midwestern American accents when she returned.
.. staying at the hotel last year, we decided to go again.
There is no subject (a word like "we") in any of the options proposed for the part of the sentence up to the comma (for the clause ".. staying at the hotel last year").
This means the two clauses, the one before and the one after the comma, are linked very hard. So hard that one of these clauses cannot be used on its own as a separate sentence. Let's check:
We decided to go again. (clause 2)
It is unclear where we decided to go exactly, but this is still a valid sentence. This means it's an independent clause.
Enjoying/Having enjoyed/Had enjoyed/Had been enjoying staying at the hotel last year. (clause 1)
When we apply any of the 4 options, the sentence does not become valid (on its own, without clause 2 attached). It could become valid if we add a subject like "we" in some of the options.
For example, the whole sentence could become valid with option c if we imagine an omitted we in clause 1 and use a semicolon instead of the comma:
(We) had enjoyed staying at the hotel last year; we decided to go again. (option c with some modifications)
But since there's no subject like "we" and there's a comma, it's much more likely that clause 1 is meant to be dependent upon clause 2: it should merely add to the meaning of clause 2, not be a standalone ("independent") clause.
Option d would not work for the same reason, moreover, it uses the Past Perfect Progressive: there should be a mention of a past event after which your "enjoyment" stopped.
This leaves us with two options, a and b. Each of these options uses a non-finite verb, hence each of this options makes clause 1 a non-finite clause.
Enjoying staying at the hotel last year, we decided to go again.
We could imagine a family making a decision while being in the process of enjoyment, but the sentence looks strange for some reason. Maybe a native speaker will explain. Maybe a non-finite clause is insufficient to carry the sense "while we were enjoying".
Having enjoyed staying at the hotel last year, we decided to go again.
This sentence is right on the spot: the non-finite clause explains the reason why we decided to go to the hotel again. To be more precise, clause 1 is what is called a participial clause or participle clause.
Best Answer
long, black, leather is correct.
Adjectives are always in the following order: opinion, size, age, shape, color, origin, material, purpose.
No one really knows WHY we do it this way. Most people don't even realize they're doing it and if you asked them if there was a rule, they'd say "no". I don't think it's even technically a "rule". But everyone does it this way and if you don't, it will sound weird
The only real exception is when one of the adjectives is actually part of the name of the object, as in a Great Dane.
A green Great Dane and a great Green Dane are two very different things. One is a jealous dog and the other is a really good politician from Denmark who cares about the environment