When you remove the double-pole breaker, it is going to open up two slots. You can replace it with two single-pole breakers and leave one as a spare, or use a special plug that is made to fill the gap properly. You can just use a conventional 15 amp breaker of a compatible brand, nothing special about it.
Your technique for changing the breaker is fine, but personally I would put the breaker into the panel first before inserting the black wire and tightening the lug. Just make sure that the breaker is turned off, or even better, kill the main to the panel if you can. I find that it is easier to tighten the lug properly if the breaker is installed in the panel rather than trying to do it in my hand, but either method will work. The white and the bare copper wires do not need to be touched.
Your technique for changing the breaker is fine, but personally I would put the breaker into the panel first before inserting the black wire and tightening the lug. Just make sure that the breaker is turned off, or even better, kill the main to the panel if you can. I find that it is easier to tighten the lug properly if the breaker is installed in the panel rather than trying to do it in my hand, but either method will work. The white and the bare copper wires do not need to be touched.