My bike has a strange problem involving the lateral stand

I have a pretty old Kawasaki ZZR 250 and it has a strange "starting" problem that manifested over the last two years:

If it’s hot outside, in summer season for example, the bike has no problem; but during winter season: when I put the keys in and I turn on the bike the engine starts well, no problem there, so I hop on the bike, lift the lateral motorcycle stand and put the first gear, still no problem. But then, when I release the clutch and I give gas the bike won’t move! The tachometer behaves fine when I don’t release the clutch but when I do the tachometer drops, and if I don’t pull the clutch in time the bike turns off. So every time I turn the bike on I have to perform several tries before the bike finally starts moving, but sometimes the bike needs like 15 minutes to finally move! Seems like I need to "warm it" or something.

But there is another strange detail: When the bike finally starts it does not have any problem, it goes on just fine, let’s say I drive the bike for one hour, everything fine, but then let’s say than I have to stop to pick up something from a shop, so I don’t turn off the bike, I simply put it in neutral, with the engine going, and I put it on the lateral stand and hop off; when I hop back on and lift the stand the bike won’t move again!! It has the same problem that it has when I turn it off from cold! So this problem surely has something to do with the lateral stand! But I don’t have a clue on what the problem actually is!

Could it be the safe mechanism to prevent me from staring with the stand down to blame?

What is going on? How can I fix this problem?