Well, for some of what you are doing you'll probably need more than just vcftools. And it does probably help to bgzip and tabix the file.
A lot of the raw stats you can get with vcf-stat, although I would highly recommend using bcftools stat instead as it is much quicker (uses the fast htslib c-library). That will give you a whole bunch of aggregated statistics on your vcf file. I'll go through each of your points:
1: What sort of information on column header? A VCF file has pretty standard columns with the only real variable being the columns for individual samples.
2: bcftools stats will give you this
3:bcftools stats will give you this
4: All of the samples in the vcf file should be the same human genome build, and this info should be in the header. Since vcf file rows are laid out by coordinates, and it is the genomic coordinates that tend to change between builds, a file where samples were not aligned and called to the same reference build would be useless and lead to all sorts of downstream analysis problems. If you could even create such a file in the first place. So you shouldn't need a special tool for this, just check the vcf file header. For the depth of sequencing what do you want to measure? Just depth per sample at each position that is present in the vcf file? Depending on the process used to create the VCF file this COULD be every position in the genome, but more likely only includes positions where at least one sample had a variant allele. If so there is a DP field in the genotypes string per sample. It isn't the total reads but is a count of the reads covering that position that passed filters (as defined by whatever program was used for genotype/variant calling). Depending on what you want, if you have access to the BAM files you may want to look at samtools, GATK, bedtools, etc for calculating coverage statistics.
5: For this I guess you would want to just write a small script, either in perl (so you can sue the VCF.pm library from vcftools) or python (using PyVCF) or whatever language you chose to go through all of the positions in the VCF file and output locations based on passing specified criteria.
6:Frequency distribution of what specifically? I am assuming you mean something more like variant type or subtype? Transitions, transversions, multi-allelic? specific substitution patterns? For some of this bcftools stats will give you raw counts, you can take the total output and send it to a file and then pass it to plot-vcfstats to get some visualizations. You could also write a script to do some secondary work on the output stats if it isn't produced/covered by the plots.