HUD/cluster light issue on cold days 89 S13

I'm having somewhat of an issue with my gauges, it only does this in cold weather, i have an 89 coupe 5spd, it has the HUD, anyways, when I go to start it on very cold days, my gauge cluster lights and HUD won't light up, the tach, temp, gas gauge still work but the HUD will not turn on and the gauge lights won't either, then after the car warms up the the Dash lights and HUD come on and work as they should, anyone have this type of issue?
On a separet note, I also have a speedometer issue, also on cold days my speedometer will be reading like if I'm driving the car when I'm parked just warming up the car, then it fluctuates as the idle changes, on this issue I'm leaning towards the speed sensor since my odometer and trip Are also moving as if I'm driving also, after the car is fully warmed up everything is working as it should on both issues.

I was reading on another forum where someone was having an issue with the dash lights, another person told him there was a small white box in the driver side kick panel that may cause dash light problems, is this true and do you guys know if this will help the first issue I stated above? Also would a bad dimmer switch cause this also?

I've had the second issue since 2007 I think after I replaced my clutch, first issue I stated only did it once last winter but now it's done it almost every cold start.

I appreciate any help/info you guys can give, thanks for reading!