You can buy AppleCare any time during the first year warranty period, so you don't have to buy it at time of purchase.
You can check your remaining warranty and/or AppleCare coverage here
AppleCare will extend the 1 year warranty for an additional 2 years, for a total of 3 years coverage from the date of your Mac purchase.
It also extends telephone support from 90 days, which is included with your original warranty, to a total of 3 years.
You cannot buy AppleCare again or renew it once it expires.
Neither the Apple Warranty nor AppleCare will cover damage from accidents, spills, etc. They only cover manufacturing defects.
AppleCare+, which provides some coverage for accidental damage, is only available for the iPhone, not for Apple computers.
Neither the Apple Warranty nor AppleCare will cover batteries that have worn out. They only cover defective batteries.
As to whether AppleCare is worth it or not, that's a matter of opinion. You'll find lots of opinions on both sides, with roughly 75% saying it's worth it. You really need to decide if it's worth it to you. If you want more information, you can search the forum, where you'll find dozens, if not hundreds of threads asking "is AppleCare worth it?" The overall consensus seems to be about 75% in favor of it.
Note that accidental damage/theft outside the insured house/apartment is usually still covered, however only up to a fraction of the total insured property value. Contact your insurance and make sure that the coverage is sufficient for your purposes.
I can see two reasons why one would want a separate insurance for the laptop:
1. a high deductible on the home insurance
2. ability to claim the laptop damage/loss without risking a premium increase on the home insurance