1996 Toyota Corolla Coolant a bit red with Transmission Oil

Took the car for oil change; Report given was that coolant is red in color Took the car for oil change; Report given was that coolant is red in color and
it requires flushing. When I parked the car in the garage, I found lot of oil leak
on the floor and the oil level was low. Took it back to the shop and they 'fixed' it.
No reason was given how it happened. I got the transmission flushed and power steering
as per recommendation. The radiator is a new one (approx 5 months).
During the next oil change with the same shop, they recommended coolant flush and mentioned
leak in their report (transmission leaking into the coolant). I took the vehicle home
and the next day found the floor full of engine oil with 2-3 drops of transmission oil
(red/pink). The vehicle was parked on level ground. Checked the oil and found it to be low.
Took it back to the same shop and complained. They inspected and took me to show the bottom
part of the car. Oil gasket and transmission gasket were leaking is what they told me.
He said the oil gasket can leak any time and that is what has caused the leak. But upon
questioning why it happened ONLY after the oil change and NEVER in between the oil change
he could not give a convincing reply. Also, when I told him the oil level was low, he
pretened as if he was not aware of it and brought the vehicle down and showed the oil
level and it was full. I am sure he had filled it and I am 100% sure it was low when I
checked it. For that matter I check the oil level once I cross the 2000 mile mark after
every oil change every few days.

I got my transmission flushed a month ago from a different reputed shop
but not sure if they replaced the gasket. I will follow up with them.

My question is how do I fix the transmission leak? Can transmission oil get into the
engine oil? The engine oil that was on the floor was 95% brown with 2-3 drops of
red/pink tranny oil.

In between the oil change there was NEVER any drop of oil on the floor.

Please advise. I feel I am being given wrong diagnosis. My thermostat never went up and
was replaced 8 months ago. I don't find anything different / wrong with the engine
behavior / accelaration.

I am now worried out transmission leak more than coolant. I am no pro and you experts
advise.

Thanks