not receiving a bid the first time you rush

You go to all the houses on the first day, and at the end you rank the ones you wanna drop and the sororities cut girls too. The next day whoever doesn't cut you, you go back to, so no it doesn't look bad cuz the girls in the sorority don't know which houses you're going back to

Final pref round u can put down three sororities max. You are guaranteed one of those sorority bids. However if too many have dropped or u only want to put one or three on final pref then u are not guaranteed a bid

You don't "rush" one specific sorority like fraternity rush. You go to every house on the first day and from then on its a mutual selection process. As mentioned already, on pref night you rank the three (sometimes fewer) houses based on the one you must want. If you go to three houses on pref night and rank all of them (aka don't "suicide") then you'll get a bid