Captain reported an airspeed malfunction and other associated malfunctions related to the aircraft air data system after takeoff in IMC. The Captain diverted and landed safely.

Narrative

On an IFR flight from ZZZ to ZZZ1; while on departure roll the copilot called no airspeed on his side. We aborted the takeoff; and as we were rolling out to taxi way advised tower we had an airspeed issue. While taxiing off the runway we noticed that the airspeed came back alive. The pitot tube cover had been damaged on a previous flight and was not installed while it had been sitting on the ground for a few days with rain in the area; so assumed maybe some water had been in the line and was not free; as the airspeed seemed to be responding normal on the taxi. We taxied back to attempt another takeoff and briefed on the callouts and the plane to go to ramp if any issues. ZZZ is not our home base and we had no maintenance at this facility. We were cleared for takeoff and proceeded with departure roll. Everything was normal and normal speeds; we completed takeoff and during after takeoff checks the right IAS stopped at 167 knots. At this time called for checklist and proceeded to go to Air Data Computer-X which was working fine. I then decided that it would not be wise to continue as we had multiple legs that day and weather at the initial destination ZZZ1 and would divert to TPA. Was a short flight and started to setup for an approach into Tampa. The weather was broken 800 with 10 miles visibility and could tell it was clearing as I could see the airport through the clouds some way out. We were cleared for the ILS 1L. During this time and slowing; a loud gear horn came on while I was at 200 knots. I advanced the throttles and I looked again to make sure ground speed was fine via GPS and it was. The Horn came on again and since we were below gear speed decided to extend the gear. During all this was responding to ATC; who noticing they gave us a tight turn to close to arriving at the Final Approach fix and they gave me a left turn to have a better intercept. I was focusing on the airspeed and the turning back to final as the ATC correction and wind from the NE...I had overshot final. Seeing the ground below; I assumed I would transition to visual soon and could continue trouble shooting as I would have to advise maintenance on the ground of what I saw. the copilot was running the checks; but he was also starting APU and busy as well..and never gave me any altitude callouts. I noticed now and ATC advised we were below the minimum altitude for that area. As I looked down I noticed we were 500 below the published altitude. I started to correct lateral and vertical and got back on the approach and had intercepted. I started coming down and was trying not to overcorrect; so I remained a little left of course until 3 mile final; but had visual by this time. Contributing factors were that I wanted to get the mission completed and the facility we were at had no maintenance; so I decided to attempt the flight as having XX years I have seen this airspeed problem before. The copilot did not have a lot of time in the last 90 days in the airplane and like me after we had the problem and decided to divert; we were focusing to much attention to the problem; which we had secured via the checklist and not enough on the approach to the divert. Seeing almost marginal weather and flown to the divert airport for years; I had it in my head this would be a visual approach. Realizing the altitude deviation I immediately kicked back into IFR and flying the plane rather than trouble shooting the horn/airspeed issue. The flight was also very short at 20 minutes; so things were going pretty fast; task saturation. I should have stopped the trouble shooting before we contacted the ATC Final; and only focused on the checklist and approach. We debriefed as a crew and discussed with manager