My tires look new but I just bought the car and I don't know for sure. They all look good except one rear tire which has cracking between the treads but nowhere else. Is this an accident waiting to happen?
Cracking is... not good. Usually it's a sign of a tire that's either been stored incorrectly or more likely pure age.
Get it changed - tires aren't something to take chances on.
It's probably worth checking the age of the other tires while you are at it. You can work out the age of a tire by looking at the DOT code - this should be a 10-12 digit code (usually prefaced with "DOT") stamped on the sidewall of each tire. If the tire was made after 2000 then the last four digits of this will tell you the year and week the tire was made. e.g. 1603 would be the 16th week of 2003.
Really if your tires are more than 10 years old they should be replaced, regardless of whether they are showing any visible signs of wear or cracking.
Not with modern tires, at speeds under 90mph.
You are safe even if nylon or steel is showing, but use your head, and be more careful if you have older tires