There does not seem to be much difference. Recombination introduces genetic diversity. Single Crossing over results in genetic recombination. Double crossovers may or may not result in genetic recombination.

No. A mutation can change an allele into any other allele. Homologous recombination can only change an allele to the allele of the homologous chromosome.

Synapsis is the pairing of homologous chromosomes in Prophase I of Meiosis. Homologous recombination (crossing over) follows.

Crossing over or homologous recombination