Translation:Do you want to live in such an apartment?
In this situation, "takovém" and "takovémto" can both be used with really small change in meaning.
You would probably use "Takovémto" if you and the person you are speaking to were in the "bytě" at the moment.
But even native speakers usually don't see any difference.
"Takovém" is more common nowadays as it's shorter and easier to pronounce. And "takovémto" might in some situations even sound a little bit too archaic or literate.
I am native AmE. While I'm quite sure it's used often, at least in the US, it strikes me as fairly colloquial, and it's not accepted at this time. But we do have several reports for variations on that theme, and it could be viewed as a declarative question, so we may decide to add it in the future.
----- UPDATE -----
We have decided to add the declarative question versions. They may take a little time to make their way through the system. Thanks for asking.
I find in oral exercises containing "v" that it is pretty inaudible unless I play it slowly. Am I right in thinking that native Czech speakers automatically deduce that it must be there because of the word endings that make up the complete following phrase? It seems to be quite a different situation to English where "in", "at" etc are generally spoken with some emphasis.
1) Due to voice assimilation, "v" is pronounced as /f/ before any voiceless consonant (p, t, k, ch, s, š, č) and before vowels. That may make it seem less audible if you're not used to voice assimilation.
2) Obviously, a single consonant (such as in Czech prepositions "v", "s", "z") is less clearly audible than a combination of a vowel+consonant (such as in English prepositions "in", "at", "of'"). The appropriate case ending, however, together with the preposition makes it clear: compare "Prague" and "in Prague" with "Praha" and "v Praze" - pronounced /fpraze/
3) As VladaFu said, prepositions (unless they're two syllables or longer) merge with the following word, i.e. they're not pronounced as separate words, which may contribute to the apparent inaudibility. "v takovém" -> /ftakovém/.