We could write a whole song between us!
The first person gives a 4 line chorus of whatever rhythm they choose, and the first line or two of a verse.
Start off simple (4 line chorus, 4 line verses), and if it works then we can get complicated (8 line verse, 4 line chorus, 3 line linkage, 4 line chorus, 12 line extended verse). Completely up to the first poster, of course.
As for time signatures - quite simply if it's 4/4, then it's four beats in a "bar". 3/4 is 3 beats in a bar. We could go into things like 3/8, 7/8 or even 16/12, but that gets very complicated for those among us who don't know much about time signatures.
It might be best to say "fast song", "slow song" etc. instead of a time signature.
You start, Tal, as it's your thread. Give whatever "rule" you want in the post, and let's see how it goes from there.