I think the issue here pingu is that you are upset and hurt bu your ex's actions, justifiably, so anything you do about this situation is based on feeling hurt, rather than real concern or worry for your DD.
I doubt a 6 month old will be confused by someone else being with her dad and then leaving if the relationship doesn't last.
I do think a person can look after their 6 month old's needs, give it lots of attention, and still work on other relationships and stay at other people's houses. I often stayed the night at friend's houses when DS was a baby, it meant I could chill out with them, watch a DVD, have a nice meal, and look after DS. At that age they sleep anywhere, I don't think it will confuse her.
I think, as hard as it may be (but from your other posts you definately have your head screwed on properly!) your ex may be right, you need to let him trust his judgement. If you have concerns that he may not meet your DD's every need, then be worried, but if he is a loving caring father with genuine concern for his DD's well being you need to start trusting him with her. AFter all, presumably he wants to be a part of her life so you've got to get used to passing her over to him and letting him decide what is and isn't appropriate.
Sorry for playing devil's advocate, I would feel the same way as you in this situation, but the feelings are from your own hurt and feelings of rejection right? So maybe the best thing would be not to react. You offered an olive branch, now bite your tongue and be relly reasonable. And set down some groundrules now, just make sure they are based on what is best for DD XX