Can a cleaved plasmid "close" without exact matching "sticky ends"? Say if you have a multiple restriction site and use two restriction enzymes on it to cleave a plasmid, can it recombine with ligase? My concern is that without the small sequence of nucleotides that were in between the two restriction sites, the two ends of the plasmid won't stick and anneal. Would it work if the two sticky ends are almost the same in sequence – two bp off instead of all five?

The plasmid won't religate efficiently if the ends are not compatible. However, it's possible that very rarely your plasmids may re-ligate, perhaps because of a tiny level of contamination with exonuclease that trims back the ends to be blunt. Or there may be spontaneous blunt breaks or other strange events. This may happen in a tiny fraction of the plasmids, one in millions, but since your transformation step selects for circularized plasmid very efficiently you may end up detecting it.

Note that these rescued plasmids are often aberrant since they are unpredictably trimmed at the ends or otherwise followed a non-designed path to circularize; this is not something you want to count on for your cloning, it's an annoying byproduct that you want to avoid.