Nath,
I can feel your frustration, especially with a bike you have just purchased (been there, done that).

Here's my 2 pence worth:-
1. Ask the guy who sold it to you about the starting technique that he used, including why he did whatever it was that he did.
Also, find out the history of the bike from him; when did this problem start to occur?

2. Get a manual for the zzr600; you will know that these bikes were/are pretty common so there should be plenty for sale, on ebay for example.

3. From the manual you will find out if it has a fuel pump; not all bikes with carbs have one - I suspect the zzr600 doesn't have one. i.e. the fuel feeds to the carbs by gravity.

4. What you describe does appear to be a fuel starvation problem; that can be caused by a range of things, such as a gummed up fuel filter, gummed up carbs, a wrong float level on the carbs, a dirty air filter would cause the engine to run "rich" but lack of a correct air filter could cause it to run "lean".
There may be a problem with the actual choke itself.
There again, the jets in the carbs could be wrong, or worn, or the slide settings on the carbs may be wrong.
So, check everything about the fuel supply that is easy to check first, so that it is all within specification, before taking apart carburettors (for example).

5. What do the spark plugs look like at the tips? Running rich or lean, or "just right"? There was a thread about this in the Yam tech forum recently which was, basically, generic to any bike.

There's a few ideas.

__________________
Dave