i know where to catch them but i would like sone more info on their drop rate,catch rate

Well they are very rare so you’d probably have to farm for a while. The drop rate is probably 1/200 and catch rate, well that depends on what level you’re monsters are in your team. If you’ve beaten the game and have a really strong team then probably 93-97%. 

is it 1 out of 200 battles or what?

0.5% chance for them to appear, probably about 96% catch rest depending on your level.

Very rare indeed.

Though there is a counter that guarantees a rare monster after a specific number of battles…I think

I think its 1 out of 300
I taped one field over 300 times and i didn’t get a dragon hatching

Think about it this way:

You flip a digital coin. 50% chance to get heads and 50% chance to get tails.

It’s entirely possible to get ten heads in a row. Or twenty. Or even thirty. It’s 50/50, after all. 

Now, pretend the chance of getting a hatchling is the tails side. So it’s a 0.5% chance to hit tails, 99.5% to hit heads.

You’re going to be hitting heads a lot AKA not getting anything but bluechicks, seaquills, and breezelings.

It’s possible to keep getting that 99.5% for a long, long, long long time. That’s why they have a counter, I think, so EXTREMELY unlucky people won’t farm one million battles, get frustrated, and throw their iPod in their fish tank.