Learn English – Refreshments to follow

In OP's first example there's an implied your (or you) after mind...

1: “Do you mind [your] closing the window?”
(or)
2: “Do you mind [you] closing the window?”

...corresponding to...

3: “Do you mind my closing the window?”
4: “Do you mind me closing the window?”

Note that 4 has becoming increasingly common over recent decades, particularly in BrE. Syntactically, everything after the word mind is a noun phrase. As a general principle, using the possessive form (my, your, his) in such contexts is "dated", but it survives more in "Do you mind [NP]" because it's a relatively formal construction anyway (and people tend to stick to older forms in formal contexts).

Also note that we tend not to use mind in this sense unless it's part of a question ("Does your husband mind our/us meeting like this?") or a negation ("I don't mind your/you phoning so late at night"). That's to say although "I mind your/you smoking while I'm eating" is grammatically "valid", most native speakers simply wouldn't say it.

Let's consider the more general context and introduce another syntactically similar verb...

5: I like ice-cream
6: I like your kissing me
7: I like you kissing me
8: I don't mind your kissing me
9: I don't mind you kissing me
10: He doesn't mind our meeting like this
11: He doesn't mind us meeting like this

I've included 5 there as a clearer example of a simple "noun phrase". Although currently most native speakers (particularly, AmE) still use possessive my in OP's example (because semantically as well as grammatically, it's a formal usage), they're much more likely to use the accusative/object forms in 7/9/11 above, rather than 6/8/10.

In light of this ongoing shift away from the possessive in constructions of this general form, I think it's probably more useful to interpret OP's first sentence as a "cut-down" version of my example 2, and the second one as a slightly dated version of my example 4. You'll still encounter the possessive in various contexts, but I don't think you ever need to produce it yourself.

EDIT: In the interests of "balance", it's well worth considering StoneyB's answer to a very closely-related question, where the final "advice to learners" paragraph recommends using the possessive. I'm not going to disagree with that, provided you wish to master "formal" usage, and bearing in mind that it would be slightly "dated, odd" to use the possessive in informal contexts such as example 6 above.

If I understand correctly, you are looking for a way to phrase the sentence that will follow the one you gave under context.

There are two issues that should be addressed with these phrases:

1. The phrase 'same issue' - are you talking about the issue your answer refers to, the issue that you are currently dealing with in your answer? If yes, I would use the phrase the issue at hand

at hand (formal)

c) needing to be dealt with now: Peter turned his attention to the task at hand. (LDOCE)

2. The potential mix up of various uses of the nouns way and path

   • they can be used as synonyms meaning:

a road, path, direction etc that you take in order to get to a particular place (LDOCE)

which is not the meaning you are looking for;

• 'way' can be used meaning:

a method that you use to do or achieve something:

In this sense it is used: The way to solve/resolve the issue. As in: There can be 2 ways to resolve the issue of energy shortage: finding an alternative source or reducing consumption.

You do not follow a way (this is the problem with your option #2). A way that you follow is usually a physical one (just follow the way until you reach the next crossroad, than turn left). (!There are exceptions to this, but they don't apply here.)

If there are several ways to do something, you can choose one. Although at first your option #1 sounded a bit ambiguous (the way we chose - is it referring to the way that you chose or the choosing strategy you applied?), in this context it is clear what you mean, so this option is fine.

• 'path' can mean:

A course of action or way of achieving a specified result

The course or direction in which a person or thing is moving (ODO)

This is often used to describe someones career path for example.It is more idiomatic to say that you follow a path than a way in this sense. Again, if there are several possible paths, you choose one of those.

You might say something like:

The answer to this question will depend on the choice of the path that we would follow to resolve the issue at hand.

While this is a grammatically correct sentence, and there are examples of usage to follow a path in various texts (in social science, physics, law etc.) I am not sure this is the best (most natural sounding) way to put it. That is, it may not be a bad sentence, but it may not be a perfect one either.

Having considered all this...

Essentially it seams that your examples convey the same meaning as the sentence you provided in the part you marked as context:

The answer to this question = the approaches we will offer

the way we view the situation => the choice of the way/path to solve this problem

way/path = approach

So, unless there is a need to say this twice, the sentence: Depending on the way we view the situation, we will offer radically different approaches to this problem. is by far the best one.