C182 pilot reported experiencing inflight icing in cruise resulting in temporary loss of aircraft control and a CFTT event. Pilot was able to land safely.

Narrative

Myself and another instructor had just finished flying and departed from ZZZ headed for ZZZ1. The other instructor was flying the left seat as PIC but I had more experience in the aircraft. We departed around XA00 with the intention of making it to clear skies on the other side of the mountains. As we were just starting to make it into the mountains; our engine made a funny noise and we began to investigate it. Carb Heat was immediately applied and no change was detected except for when we turned the carb heat off. The engine would run rough for just a second and then would return to normal. After adjusting the mixture to a richer setting; the roughness from the carb heat being turned off went away. By this time; the sun had just finished setting and we were losing our visual reference due to a solid overcast layer that dominated the region. We ran into some unforcasted precip at 7;500ft while we were outside of the clouds. A very small amount of ice had accumulated on the wings and we adjusted course to clearer skies to get out of the precip. We asked center if they had any precip on radar; and they advised that they could not see any where we were at. Due to deteriorating conditions; we elected to terminate the flight immediately and found the best weather we could find near us. This led us to ZZZ2 airport which was reporting VFR conditions at the time we selected it. We called center and advised that we would need a pop up IFR and we would be changing destination to ZZZ2. We asked for the RNAV approach and center vectored us for the RNAV XX and gave us a descent instruction to 5;000 feet. I knew this would put us in the overcast layer and we would begin to pick up ice. I advised center that we would not be descending and remaining at 7;500 to limit our exposure to the ice. Center gave us a block altitude and we complied. Center cleared us for the approach and I waited until ZZZZZ to begin my descent to limit my exposure to the ice. As I was flying from the right side of the aircraft; I had a very hard time seeing the HSI and I was all over the place down final. The weather had deteriorated beyond the weather that we had originally picked up. At points on the approach; I had VSI indications of around 2;000ft per minute. I did receive a terrain warning on the Garmin 430 and corrected it appropriately. As we came out of the clouds; I believe I came within 150 ft. of a mountain. I busted the minimums on the approach and the other instructor found the runway. Once I had the runway in site; I pulled the power and dumped the flaps. I possibly oversped the flaps as well. I slipped the aircraft all the way down to find a snow-covered runway. We were able to land without incident and canceled our IFR flight plan. After debriefing myself and the other instructor; we realize that we are very lucky to be alive. We should have never departed ZZZ and waited for the following day for better visibility. I believed that I was invulnerable and was quickly humbled. I hope I never have to experience anything like that again.