1 indicates that you are responsible for some mistakes.
2 Isn't using the phrase "for my part". It indicates that for your "part" (perhaps a script for a play) there were some mistakes made. Perhaps they got you the wrong costume.
3 Doesn't make sense.
4 Is fine, but cumbersome. A native English speaker would probably say "If you cook a meal, then I'll do my part by doing the dishes."
"on my part" is usually used to indicate something that proceeds from your actions [in a group of other actions possibly by other people], rather than to indicate you performed a specific action yourself. In your example you are saying that you are responsible for a few mistakes.
"for my part" is used to indicate that something forms your share of some set of actions. For my part I'll drive you over, if you do your part by paying for the fuel.