I think you can't do (2) and (3) without the mother.
(1) is doable to a very high accuracy. It is better to start with fastq. You need to generate a multi-sample VCF first. Here is a high-level overview: a) align reads with bwa-mem; b) sort alignments with samtools and then index the resulting BAM; c) call variants jointly with freebayes (or GATK if you like challenges). After you get the two-sample VCF, you go through every site where the father is homozygous and it is a high-quality site in both samples (e.g. average coverage, high mapping quality, not in difficult regions, etc). Among these sites, check if the child has the paternal allele. If you have a father-son pair, the child is expected to have all the paternal alleles in theory. In practice, if >99% of paternal homozygotes are present in the child, the two samples will almost certainly be a father-son pair. The actual threshold may depend on how you define "high-quality" sites. Errors may lead to apparent mendelian violations.
In principle, it is possible to do a similar analysis with the two provided VCFs, but you need to be very careful about various technical artifacts.