Cannot understand the grammar used in the sentence "I'd love to have lived in the 19th century"

Question

I came across this sentence in the one of the English knowledge tests, and it really confused me. It turned out that I cannot say "I'd love to live in the 19th century" because of some grammar rules related to the past tense.

I searched about this, but all I got were only more questions and misundestanding. Some sources said that I can use the "I would have loved" construction, but I'm not sure whether I can use it in this case.

So, which of these examples are correct and what sense do they convey?

1. I would love to have lived in the 19th century
2. I would love to live in the 19th century
3. I would have loved to have lived in the 19th century
4. I would have loved to live in the 19th century

And what about continuous tense?

5. I would love to be living in the 19th century
6. I would love to have been living in the 19th century

Answer 1

There are two ways of looking at this:

#4 expressing a past irrealis

I would have loved to live in the 19th century

#1 expressing a present wish about a past situation

I would love to have lived in the 19th century

...and possibly a third suggestion:

#3 expressing a past wish describing the sustained action of living out your years during the 19th century.

I would have loved to have lived in the 19th century

Answer 2

You can't logically say I'd love to live in the 19th century, because it makes it sound like a future possibility.

I would love to have lived in the 19th century - I would love to have had that experience.

I would have loved to live in the 19th century - if I had been alive then I would have enjoyed the experience.

Since time travel is not a reality, both sentences in effect mean the same.