I waited about four months before using my cup. I bought a mooncup and love it! I'm actually looking into buying sponge tampons, too....but don't know how they'll interact with the string. My suggestion is to make sure the string is up by your cervix in case you are concerned about tugging too hard and pulling it loose.

I have no personal experience, but for what it's worth, I have a relative who has an IUD and uses a DivaCup, so I suppose it's just as subjective as without.  Some people prefer Adidas to Nike; some people prefer Lunette to Diva.

I wouldn't use a cup immediately after having your IUD inserted, but after the inital bleed, you should be fine to start using any menstrual cup.  
Often IUD strings work themselves out of the way, if they are long enough, the most common place to find them is actually wrapped around the cervix.  Regardless of their position, it usually takes a pretty significant tug on the strings to pull an IUD out, and that shouldn't be happening with your cup.

All the best!

I agree with Holly - let your body recupe after having your IUD inserted.  You'll be a little tender.  Be sure to tell your physician that you use a menstrual cup so she/he can cut the strings shorter.  This will ensure the cup will not dislocate your IUD.  As far as what brand, they all offer unique features.  For Lunette, the cup rim is less pronounced than other brands which could offer more comfort.  Hope you check it out