All this sounds really normal and healthy, christiana
You do not need to pat a baby on the back, especially not for ages. This is just something we do 'cos we have always done it! There is no real physiological effect - putting a baby upright can sometimes coincide with a burp but it could be coincidence
It sounds like your baby manages to cope well with her burps and farts and can safely 'permitted' to do this. Sleeping on you is normal and healthy and having your hands free to get on with other things is made easier with some sort of sling.