CRJ-200 Captain reported multiple system failures during approach. Flight crew diverted to alternate airport for an uneventful landing.

Narrative

While at 10;000' enroute from ZZZ1 to ZZZ; an erroneous indication from the radio altimeter on the CA side was identified. It read as 0 feet AGL. The aircraft was in icing conditions with the ice detection probes actively detecting ice build-up on the aircraft. All anti-ice equipment was functioning normally. After completion of a QRH procedure; no further action was required as only one RA had been identified as failed. As we began to receive vectors and lower altitudes from ZZZ approach for ILS XXC; the gear horn had began to go off. We determined that the aircraft was giving us this aural warning due to the radio altimeter malfunctioning. The horn was ignored; but it could not be turned off. Once on the approach just prior to ZZZZZ at 5000'; approach mode was armed and the aircraft captured the localizer. Later; my PM noted that we were about half a dot above the glide slope and that the aircraft had not intercepted it. I immediately disconnected the autopilot; deployed some spoilers to maintain the instructed speed as I began a descent rate of 1500' FPM to attempt to re-capture the glide slope. At about 4500'; a 'SINK RATE' aural warning was given with almost immediate subsequent warnings of 'CAUTION TERRAIN'; 'TOO LOW'; 'PULL UP'. Per our standard operating procedures; we executed an EGPWS (Enhanced Gond Proxy Warning System) escape maneuver because we were in IMC. As soon as the climb was initiated; the aural warnings ceased and my PM notified ATC we were executing the missed approach procedure. ATC requested we track the localizer and descend to 4000'. Autopilot was re-engaged and the aircraft was now leveled off with flaps and gear up at 5000'. Upon initiating descent to 4000'; the aural began again and we asked ATC to verify our altitude. ATC confirmed our altimeters were correct; so we determined that our EGPWS was giving us erroneous warnings that should be ignored for now. Seconds later; the 777 aircraft in front of us was also executing a missed approach. Around 0.5 miles remaining to the airport; we received a vector off the published missed approach course to continue West and coordinate with ZZZ approach. After swapping frequencies; we were notified that ZZZ was adjusting its runways and swapping to an East configuration now. The ATIS I remember at this point was roughly 020/26G35 with visibility at 1 1/4 and blowing snow. I believe the RCCs (Runway Condition Code) were now 4/4/4 at this point for all runways. ZZZ approach attempted to put us on immediate vectors for XXC; but we said we weren't ready to conduct the approach. After being handed off to ZZZ departure; my FO and I determined it was safer to now divert to ZZZ2 - our planned alternate per our flight release. While we could have legally conducted an approach into ZZZ with the weather conditions; it was near max crosswind with an ILS that would've been down to minimums. We could not do a CAT II because of our RA failure. In addition; we would've had EGPWS warnings during the entire descent while in IMC and icing conditions. We had already gathered weather at both ZZZ1 and ZZZ2; and the VMC conditions; closest proximity to us; and a North-facing runway are what contributed to our decision to divert there. While enroute; I handed the controls to my FO and revisited the QRH to see if there was any procedure for an EGPWS failure. Because we had no status or caution message to directly point me to a checklist; I revisited the radio altimeter failure and ran the QRH as if both RAs had failed - seeing as it mentioned a total failure of the EGPWS system (even though we didn't have the associated GPWS FAIL status message). I gathered that we might also have erroneous or a lack of TCAS alerts in addition to the possibility of ground spoilers not deploying on contact. Shortly after completing the checklist; my FO was now with ZZZ1 approach and receiving initial vectors for ILS 1. We already had discussed the weather and NOTAMs; so I completed my approach briefingand shared the QRH information with my FO. Due to the lack of possible TCAS; a failed RA; continued erroneous EGPWS notifications; and still being in IMC; I elected to Advise ATC to ensure our approach would be prioritized and increase any traffic separation. I requested that ARFF (Airport Rescue and Fire Fighting) be deployed on the runway and follow us as we come to a complete stop before electing to continue to taxi. We were in VMC around 3000' and the remainder of the approach and landing and taxi-in was uneventful. After confirming with tower and ARFF that the aircraft would be okay; they left . We unloaded passengers at the terminal.Cause: The cause of this diversion was due to the mechanical RA failure which resulted in the subsequent erroneous warnings of a gear horn and EGPWS terrain warnings. It was also due to the difficult weather conditions in ZZZ; nearly exceeding the aircraft and approach limitations. Lastly; severe icing was noted around the ZZZ airspace; contributing the need to divert away from ZZZ to ZZZ2.Suggestions: We could have immediately returned to ZZZ1 or even diverted to ZZZ2 upon the initial recognition of an RA failure instead of electing to continue to ZZZ. I was not familiar with what systems would subsequently fail after an RA failure; so I could have been better equipped with that knowledge as well to aid in that decision making.