In the the late 19th Century, the British Prime Minister was Lord Salisbury - family name, Robert Cecil. 'Bob' is, of course, the short form of 'Robert'. He gave a variety of Government positions to his not-very-able nephew, Arthur Balfour. That gave rise to 'Bob's your uncle' - meaning something pretty easy. Another possible source is from a phrase 'all is bob' meaning 'all is safe' or 'everything is well'.
Now I didn't know that, QM; I like that explanation. Re your second point, I've come across someone - a Midlander, like me - who uses the phrase "bob on" to mean "exactly right" - as most of us might say "**** on" or "dead on" or "bang on".
I've never heard 'bob on' used in that way myself, Naro. I wonder whether it is related to the bob, meaning a weight, as in a 'plumb-bob'. A plumb-line is seen as an absolute marker of exactness in building, decorating etc, so 'bob on' might - as you suggest - have come to mean 'exactly right' for that reason. 'Plumb' itself, used as an adjective, also means 'exactly right', of course.