Sounds like a sprain ...
Any injury, ice for the first 24-72 hours (they usually say 10-15 minutes at a time), you've got the rest right. Never put heat on an acute injury.
Most important thing about an ankle sprain is the rehab after it heals. Once you can do it without pain (probably 3-6 weeks depending on how severe it is), you'll want to be doing one-legged balance exercises (start on a stable surface and progress to unstable), range of motion work, foot strengthening exercises like towel skrunches. Personally I would keep it wrapped or braced during activity until you've been doing the balance exercises successfully for at least a month or two.
IANAD, etc., etc., but I've sprained my ankle way too many times because of inadequate rehab... I think if you can't put weight on it within a few more days, you might want to see your PCP to get an X-ray, but it doesn't sound broken from your description.