Upon going out side, a coat was given to me by my mom. (Is it because
the subject cannot be changed from I to a coat?)
It's in the passive form, but the only mistake I can see is 'out side' which should be 'outside.'
If you want to write it in active form...
After going outside, my mom gave me a coat.
Coming to the second example...
As soon as going outside, I put on my jacket. (I know as soon as is a
conjunction, but I don't see any wrong with this sentence.)
Here you didn't mention who's going outside because it requires to be mentioned either before or after 'as soon as.' Also, it's not an ongoing action, so you can't use 'as soon as' with 'going.'
I put on my jacket as soon as I went outside.
As soon as I walked outside, I put on my jacket.
These two are possible alternatives.