Good question. No, it doesn't refer to kidding (although that would be similar "don't fool with me" could have that meaning). "Don't make a fool out of me" means that you want someone to avoid damaging your reputation by his behavior with other people. This may refer to a situation where he is deliberately trying to make you look bad, or it might be that his or her foolish behavior ("making a fool of himself") could reflect onto you, due to your close relationship. There's usually no significant difference between "don't make a fool out of me" and "don't make a fool of me."

'Don't make a fool out of me' means don't treat me like a fool, or don't make me look like a fool. It would mean the same without the 'of'. 'Kid' has a different meaning. If you 'kid' someone, you tell them something that isn't true.

I guess there is no difference between "don't make a fool out of me" and "don't make a fool of me". "don't make a fool of me" is common. It is similar to "don't kid me" to some extent. Btw, don't kid me, rather than don't kidding me.