Is there a database of patent provisional applications available?
No. Patent applications (generally) are published after 18 months -- but since a provisional expires after 12 months, it would have to have been turned into a nonprovisional to survive to publication.
What happens to an expiring patent provisional?
It just expires.
An abandoned (expired) provisional application can only be accessed by the public if it was published (after 18 months, see above) or if it was identified in a published patent application or issued patent. See 37 CFR 1.14(a)(1)(iv).
If a patent provisional has been expired, but another company filed a quite similiar patent (either as a provisional or through a regular patent application), but only after the original filing date of the expiring patent provisional--can the other company proceed with its patent application?
Yes -- but chances are the USPTO would declare an interference, and then it would be up to the two companies to prove who invented the invention first. The problem is, in the U.S., the first person to invent is entitled to a patent, with the caveat that the first inventor must have been diligent in reducing the invention to practice. If the first inventor did reduce to practice, filed a provisional, and then abandoned said provisional, that could be an indication that the inventor was not dilligent, and instead abandoned or concealed the invention, and in said case the first inventor would no longer be entitled to the patent. But this would depend on all of the facts of the situation -- an abandoned provisional is not dispositive, its just evidence. But in an interference proceeding, an abandoned provisional may hurt you more than it helps.
Can the expiring patent provisional affect in any way other similar applications?
Generally no -- an expired provisional essentially never existed (the exception being if a nonprovisional continuation or continuation-in-part application was filed while the provisional was still alive, and then allowed to expire, then the expired provisional is still "alive" for the purposes of establishing the priority date of the nonprovisional application).