Couldn't it mean 'Do they want an apartment here'?
No. BYT (short Y)- apapartment; BÝT (long Y) - to be
am I the only one who hears this as štějí instead of chtějí ? Is this standard pronunciation?
Nope, I hear it, too. And I've been told by a Czech native speaker that it's easier to pronounce the "š" sound before "t" than the "ch" sound.