Lear Captain reports low altitude alert from TEB Tower during LOC 19 approach and exceeding missed approach altitude during the missed.

Narrative

We were proceeding to TEB via the JAIKE 2 STAR. New York Approach advised us that TEB was using the ILS 19 and the GS was out of service. The First Officer was the Pilot Flying in the left set. I was Captain (Pilot in Command) and Pilot Not Flying in the right seat. The First Officer briefed the approach including the missed approach procedure while I monitored the aircraft. As we were being vectored; Approach informed us that the last aircraft had picked up the airport 'right at minimums.' We intercepted the LOC and configured according to our standard profile for a non-precision approach. After passing the FAF; I set the altitude alerter to 600 ft MSL; which is 20 ft above the MDA for the approach. Upon reaching MDA; Tower advised us that we had triggered a low altitude alert and re-issued the altimeter setting. I verified that we were on course and inside of the FAF and that the altimeter was set correctly -- the updated altimeter setting had increased .03 inches Hg from the ATIS. I also looked at our GPWS display and verified that there was no terrain displayed on our flight path. I attribute the low altitude alert to the fact that we used a higher descent rate than a typical ILS approach so that we would be at MDA prior to the missed approach. I suspect that we went below the ILS glide path; which triggered the alert. At the missed approach; the First Officer and I could see parts of the airport; but the runway was obscured; so we executed the missed approach. The First Officer pushed the go-around button; pitched up and went to takeoff thrust. He did not call for 'Flaps 8 degrees;' so I queried him. He called 'Flaps 8 degrees' which I selected. I called 'Positive Rate' and he called 'Gear Up' which I selected. I called Tower with 'Missed Approach' and he was handed off to Approach. I noticed that the First Officer's Flight Director was in Go Around and Roll mode; so I selected FMS then NAV so that he could fly the published Missed Approach Procedure. I then called Approach with a