Can I be assured the following code generates a unique cookie id in a mod_perl environment?

Nope.

For example, it's perfectly feasible that two requests could be processed in under a second on a single Apache process, and the same random number come up twice.

Also, since you're not separating the values you use with a delimiter you can get the same string from different values, e.g.:

$ToBase62->(time)	$ToBase62->($$)	ToBase62->( rand($$))
1234	567	89
12345	678	9

both give the same ID.

Personally I tend to use Data::UUID for creating unique IDs.