Chromosome related problems.
How many versions (alleles) of a gene can we carry if every chromosome is in pair?


The phrasing "if every chromosome is in pair" sounds a little bit funny. But it probably mean "or a diploid individual".

If an individual is diploid, then it has two copies of any given locus. If the term is new to you, a locus is a position in the genome (it may correspond to a gene or not). With two copies an individual can carry maximum two different copies, that is two alleles. The answer is two.


  What happens during the gamete formation (meiosis) ? Do the gene pairs stay together?


You nailed it!


  What happens then to the paternal and maternal sets of genes that we inherited ?


This is a slightly vague question IMO but I would suppose the main point that is being expected is a description of segregation and recombination (cross-over). You answer was good but I would suggest.

"The maternally and paternally inherited haplotype will recombine (cross-over) during meiosis so that a gamete resulting from meiosis will contain a single haplotype which is a 'mixture' of the maternally inherited and paternally inherited haplotypes."