Bad? No. Why does she feel the need to make you jealous, though? She may just be needy or an attention whore...or she may be genuinely (and accurately?) feeling neglected. It's to your benefit to determine the cause and react appropriately.
Recall the story I shared with you yesterday...not my doing, but worthy.
No, just ignore all of it.
No not at all. A relationship shouldn't be about games...unless its monopoly