You can use either vcftools or bcftools. You'll just use the isec command with the -C (complement) option. Note that this is position based rather than exact variant based.

Regarding point 1.

I think you can do it with R.

try to transform your 2 vcf files into Dataframes (DF1 and DF2) and then subset the content of your chromosomic position of the DF2 containing the private variants from the one having all variants (DF1)

this is an example:

DF1<-read.delim("....\allSNPs.vcf",header=T,sep="")
DF2<-read.delim("....\private.SNPs.vcf",header=T,sep="")
singletons.DF1<-DF1[!(DF1$chromosomic.position %in% DF2$chromosomic.positon),]

I wrote a tool to include/exclude the variants in a VCF file: https://github.com/lindenb/jvarkit/wiki/VcfIn

I know this is an old question but when searching for filtering out singletons(and doubletons too) i found this post in the first results. My solution to this problem for a diploid organism, purely in bcftools is:

bcftools view -i 'COUNT(GT="alt")>1' samples.vcf

This is assuming that all singletons will have the alternate allele. If singletons can be found with the ref allele i would add the following:

bcftools view -i 'COUNT(GT="alt")>1 | COUNT(GT="ref")>1' samples.vcf

This logic can be extended further to only heterozygous singletons and such