Do ことである and ことがある mean the same thing?
I don't think so.
である is basically equivalent to
です (the copula), so
ことである can be replaced by
こと is either nominalizing the preceding expression, or is used as literal "thing/fact".
This is an important matter.
[I] have something* very important. (lit. A very important thing exists.)
*Note: unlike similar
こと refers to intangible things.
ことがある (or polite
ことがあります) can describe either an event that happened in the past (with past form verbs), or something occurring with some regularity (with non-past forms).
[I] have been to ~ several times
This usage usually describes personal experience.
It cannot be replaced by
ことです to express the same meaning.
Fact of the matter (?) is that [I] went to ~ several times*
*I'm not very sure it's a correct translation.
See also The different usages of ことがある
References: A Dictionary of Basic Japanese Grammar by Seiichi Makino, articles on aru1,koto1, koto2, koto ga aru 1, koto ga aru 2