Eat well, sleep well.
Take zinc supplements (more than the RDA)
Do 30-40 minutes easy training per day.

All of the above should make the cold run its course quicker than taking complete rest.

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Acute phase I will have a few days off as much as I hate it. Otherwise it just hangs on for longer or returns.
Symptoms are neck up then I will zone 2 workouts till I start to clear up, then ease back into it.
I was (and often still am) who hate forced rest as I swear any gains leave me by the hour lol. But it really is worth taking a couple, atleast at the start. Easy easy workouts, your body is busy doing other stuff, no fitness gains can be made here, you can only go backwards.
As for your situation your probably just having lasting effects. If workouts stay this way you prob need to dial it back to very easy zone 2 stuff.

It depends. I have had viral illnesses that I have simply reduced volume and intensity, and trained thru. I recently had an 8 day cold where I only managed two workouts. It was horrible. I felt weak and had very high HR at slow paces. Listen to your body, it will tell you what to do.

I was 8 days off my last one a few weeks ago. That's the longest I've been off without training.

Had to go to the Doc for that one before it got worse.

It is not strange at all. A cold is a viral infection and healing can take up to nine days. If you don´t cure it properly the risk of getting a bacterial infection on top is there. If you are sick don´t do sports. Quite simple.