It's probaly not blood you are tasting but an increase in the salinity level of your saliva & mucus.Spit on something white when your experience this & see if there is any blood.If there is it might be time for a doctor visit to determine the cause.
The solution is probably to just hydrate better before the run.
The cold changes the viscosity of the mucus & sometimes increases the quantity as well.This makes you more aware of its taste
Now for the running.It is a high impact form of exercise.Hard on knee & hip joints.You may pay the hard way for it when you get older with hip or knee replacement.So slow it down to a brisk walk.Or swim if you have access to a pool .Much better IMO & some others opinions as well.

I guess if the quantity of blood was small enough an occult test could be done on the saliva.
For the blood tasting  salty mine does if I lick a bleeding wound but I guess that could be sweat I'm tasting.I've never tasted blood that was drawn so it wouldn't have skin sweat contamination so I can't say if it would taste salty.The same would be true for saliva or mucus contamination.
btw I'm not curiuos enough to taste drawn blood.
I disagree on the running being safe .Regardless of the shoes or surface it is still a high joint impact form of exercise.& this produces wear in the joints.It is just less impact on softer surfaces & with correct shoes.
So only run when in danger like if a bear is chasing you & then only faster than the slowest person with you.If alone play dead for a bear anyway.

You might be in front I haven't ran for a long time.I was pretty fast at one time when a lot younger & didn't consider running a bad exercise.I think I could still call on the old technic if the adrenaline was flowing.
The damage is still being done to the younger runner it just doesn't show up until older.Although not necessarily a runner but it was probably part of her exercise regimine Mary Lou Retin is now making knee replacement commercials.
Seeing a doctor might give you some peace of mind.Surely an occult blood test of the saliva & mucus wouldn't be too expensive.I guess the best time would be right after you ran in the cold.You could make an appointment so they could take the test right after you ran in the cold.That shouldn't be a problerm as doctors always get you in right at the appointment time.

Yep. light, cool, and fast. But not a damn bit of support in them. You were actually better off with a pair of converse all stars. At least they have a steel strip in them.