You're right it is your bike for you to do with what you will.
Promise one thing that you won't take an angle grinder to it and remove that dinky little bracket on the seat tube or those funny knobby bits on the down tube.
I'm sure that you will soon come around to thinking that fixies are a bit last year and the way to go is to put some gears on it along with a nice set of brakes so that you can belt downhill on it with a great big grin on your face.
Nice bike by the way you're very lucky to have it in such good condition.
Don't think those cloth mud guards will last too long, mind.