Muscles

Heat is for sore muscles.

Ice is for inflammation and pain.

So ice, not heat. 20 minutes on, 20 minutes off.

Also, you may think your shoes are great, but unless you've had a fitting, they might not be right for you. And slow your pace down.

Not an expert by any means, but I have been told both chiropractors and physical therapists that you want to AVOID heat for shin splints and apply ice instead. You should Google it to be sure, since ice seems to be the opposite of what most everyone else is saying here :-).

yes. When in doubt RICE your injury:

Rest
Ice
Compression
Elevation

Heat can actually cause some types of injuries to worsen, this is rarely the case with ice. I think growing up they used to tell us ice for 3 days and on the 4th day heat.

If you find that it's due to weak anterior tibialis you can do exercises specifically to strengthen then (handle of weighted bucket over foot and raise, move marbles from left pile to right pile and back, lay a tea towel under foot and scrinch up using foot). However, the most common cause of shin issues is shin splints and the most common cause of shin splints is old shoes with worn insoles or poorly fitting running shoes (shoe weight can also be a factor). A common cause for general running injuries is too much too fast.

Going back to shoes, I buy 2 pairs every 4 - 6 months and rotate the shoes out every other workout. I generally bring them home, take them out for a test run, and if they don't feel perfect they go back and I try out another pair. Nothing worse than new shoes that take you off the road for 6 months (trust me).