Mechanics left the cap off my radiator. Can this cause proven damaged to my engine?

I had a service done on my car
I drove my car home (40 min drive) and noticed the heating wasn’t working.
I checked under the bonnet the next day and there was no liquid in the radiator reservoir and the cap was missing.
The mechanics came out to me and filed the car with water and coolant and they constantly revved the engine, I assume to get the air out because my heating wasn’t working. When they started they just put water in it, which I dont think was good.
Eventually the heating started to work and they filled it up with coolant.
I drove it the next day and the heating and air con is not as it should be and feels blocked. The heating is on and off and for the first couple of days the water in the radiator went down a CM each day. I assume the is because it still has air in it. (its fine now and hasn’t gone down for a while)
The water the next day browny blue color, it wasn’t before, it was a clean transparent redy greeny color, I think this is how it should be.
They brought the windscreen washer with them (when then came out) and I think they may of filled the car up with it as well as the red coolant they used.
I have no viable temperature gauge in my car (because its digital) I cant tell if my engine is overheating. I am worried there is not enough coolant in it to keep the engine cool.
I have had it checked by 3 other mechanics (because I dont trust the original garage)
The first one dipped paper into the radiator to show me the water is clean, he said it just looks dirty. He said the heating issues could be because there is still air in it.
The 2nd mechanic said he bled the car to try and get the air out to try and fix the heating. He said there wasn’t much air in there and this didn’t fix the heating.
The 3rd mechanic checked it over and said the fluid is probably blue and red coolant mixed together to make it look brown. He said if there was washer in it it would crystallize and it hasn’t. He advised it should be OK and to see how it goes.

Questions…
How can I check to see if any damaged has been done to the engine, and how can I prove it? If anything happens to the engine, I would like to make sure the original garage pays for the damaged they have caused, if any.
Is mixing blue and red coolant together OK?
Do I need to flush this liquid out of the radiator reservoir and engine and start again?

Any advice would be appreciated 🙂