both (x) and (y)
1. How the structure "both (x) and (y) applies to the both existing in the following context? Is it like:
both (on to the street and beyond) and (even back into the gallery)?

2.And does the following sentence "and as Butt puts it, in terms of" refers to both (x) and (y) explaining the whole approach of the author?

3. Is "latter" referring to "the spillage of art back into the galleries"?

Context:

For my purposes, though, I wish to concentrate for the time being on art’s ‘performative spillage’, both on to the street and
beyond, and even back into the gallery, and, as Butt puts it, in terms of ‘the experience of art as a profoundly embodied experience’. Importantly, the latter occurs not only as part of culture – and, therefore, presupposes the inscription of certain constitutive values and conventions – but also, by implication, in space. Thus

Question

both (x) and (y)
1. How the structure "both (x) and (y) applies to the both existing in the following context? Is it like:
both (on to  the street and beyond) and (even back into the gallery)?

2.And does the following sentence "and as Butt puts it, in terms of" refers to both  (x) and (y) explaining the whole approach of the author?

3. Is "latter" referring to "the spillage of art back into the galleries"?

Context:

For my purposes, though, I wish to concentrate for the time being on art’s ‘performative spillage’, both on to the street and
beyond, and even back into the gallery, and, as Butt puts it, in terms of ‘the experience of art as a profoundly embodied experience’. Importantly, the latter occurs not only as part of culture – and, therefore, presupposes the inscription of certain constitutive values and conventions – but also, by implication, in space. Thus

Answer

OK, with that analysis, I can finally try to answer your questions:
1. It's ambiguous. The alternative parse would be "both (on to the street) and (beyond)". That comma after "beyond" is  confusing, and seems to end the scope of the "both", but it's necessary to nest the "and" as he wants, so he may mean what you said. It would be better rewritten, perhaps as "onto the street and beyond, as well as back into the gallery, ...".
2. Who knows. See my previous response to see my take on that.
3. It would be the second of a pair of items, typically separated by "and". (If there are more than two, it would be "last" rather than "latter".)  That would then be "in terms of ‘the experience of art as a profoundly embodied experience’", but as I mentioned, I don't that's well constructed, and he probably means the " ‘the experience of art as a profoundly embodied experience’ ".

Whew!

Answer

It's not very well written, is it? There are rather a lot of adverbial phrases and follow-on "and"s in here, and the trick with this sort of thing is to unwind them all, pull out the modifiers and see what's left. I think the sentence may actually got the author into syntactical trouble, depending on what he was trying to say, which doesn't help. If  we reduce this sentence to its more basic form, you get " I wish to concentrate on art’s ‘performative spillage’ and in terms of ‘the experience of art as a profoundly embodied experience’. (The problem here is that I don't think this makes much sense -- as far as as any of this woolly language makes sense --  and it's not clear what he was trying to attach "experience ..."  to with the final "and". The only parse that seems to work as far as I can tell is to attach it to "concentrate".)

So we pulled out "for my purposes", which is an adverbial phrase modifying "wish" and which itself is modified by "though". We also pulled out "for the time being", which modifies "concentrate". We have "both on to [or maybe "onto"?]  the street and
beyond, and even back into the gallery" as an adjective modifying "art’s ‘performative spillage’". And finally, we pulled out "as Butt puts it" modifying -- well, who knows? -- something that follows. (I'm starting to think think what might actually have been intended there is something like" ...art’s ‘performative spillage’ -- both onto the street and
beyond, and even back into the gallery --- in terms of ‘the experience of art as a profoundly embodied experience’, as Butt puts it." (Whatever that means, and using dashes to replace some commas to indicate the nesting better.)