CE-680A flight crew reported a Low Altitude Alert and Glide Slope Warning while on approach.

Narrative

ZZZ1 - ZZZ XA:20 visibility 4sm Haze; no ceiling This is my first time flying into ZZZ with hazy weather so the Captain and I briefed what might happen in cruise with the weather we had available. While we were descending to our destination and after getting the ATIS into ZZZ that advertised landing RW XX with no mention of a circle; planing and briefing for vectors for the Visual RW XX backed up by the Garmin Visual RW XX we were told to contact approach and told to expect the ILS RW XY Circle RW XX with no other instructions. I briefed the new approach to circling minimums; over fly the field and enter a left hand pattern to RW XX. Approach cleared us for the ILS not localizer RW XY; Circle RW XX; maintain to 3000ft till established. I briefed using VS and Nav to descend to circling mins but by habit Armed APPR mode at 3000ft to capture the GS. We switched to Tower when we intercepted the course; Tower then cleared us to land RW XX; to circle to the southeast; he would call our base turn and also advised us that a helicopter would be in below us to the southeast. We saw the helicopter on TCAS. I got fixated on the helicopter traffic and realized around the FAF that I needed to set MDA and use VS to descended to MDA. So I selected those modes on the AFCS to descend while Tower was calling the helicopter traffic to our right and below us. In the task saturated moment while worrying about hitting the helicopter I us the VS wheel to get down and must have selected a steep descent more than 800fpm rate because while trying to figure out where the helicopter was we got a glide slope warning and a low altitude alert from Tower at 1700ft (MDA was 1400ft). I immediately arrested the descent and we started our turn left to do a right downwind (no standard pattern) for RW XX. While on short final we received a glide slope warning and wrong runway on the FMS and Audible alert. I was still 500 stable. We both decided to continue and landed and taxied into the FBO without incident. Suggestions: In the future it would be nice to get a heads up on the airports ATIS that instead of landing RW XX; the ATIS should say they are using the ILS RW XY circle to land RW XX to eliminate the changing of plans multiple times since we both have not flown into this airport before. In hindsight if I knew the Haze was worse than I thought I would have asked approach what approach we could have expected. I also should have realized I armed the approach and should have immediately switched to NAV mode and VS to capture MDA. I also had the ceiling to stay higher and should have but not knowing how bad the haze was I went to circling mins. I should have also let the PM look for traffic and worry about using VS to descend. In my mind I was ready to have to go around because of a TA/RA with the helicopter that Tower called and was trying to give him adequate room. With everything changing so quickly I should have verified with the PM that my plan made sense but with the traffic being a threat and the many changes I failed to do that.