Figuring out if populations from two regions share the same Neanderthal haplotypes
What I am saying here is the way I would approach the problem. And as such this post is meant to give you some direction. Please do not hold me responsible if you do not get the expected results.
First off, I don't know how the Plink format works, you would do well if possible to convert that to a BED/GFF/GTF file somehow.
I looked up the file format, and most of your columns are significance scores. For a start I would use the BH FDR &amp; p-value as a preliminary filter and later use the Bonferroni corrected p-value.
Wiki link to FDR
The wiki link should also give you an idea why I say later on move onto bonferroni.
Interesting questions asked on cross validated for FDR


Provides a vague idea on FDR
Provides a relation between p-value and FDR: Re-read the answer a few times and you'll get it.
Check out the books if you want some understanding of Statistics


Using the SNP data, I need to try to find out if they have Neanderthal haplotypes in the same parts of their genome. I've found a map of putative Neanderthal introgressed haplotypes from 1000 Genomes Project's European and East Asian genomes

I looked up the file, these are regions with an average span of 73.4kb across the entire genome. So what you want to do is map the &quot;statistically significant&quot; SNPs from your plink file to these regions.
If you don't know how, check out the UCSC genome browser. A detailed step by step would be too big an answer, but the overview goes something like this;

Choose your reference genome from assembly
Create two custom tracks, here you can upload your SNP data in bed format, which is the easiest to make. Link to format And add the existing haplotype population.bed file.
Go to intersection, to create an intersection between these two tracks.

This will give you an idea as to which of your haplotype regions are present in the population.
Ofcourse, it isn't that easy, because you have to make a consideration here. Do you choose to say the population is carrying that haplotype if a single SNP is present in a region which spans 73kb? That is more than the average size of a gene in Humans.
As far as the Neanderthal and Denisovan genomes are concerned. I would again point you to the UCSC genome browser where they have the Neanderthal and Denisovan assemblies and their corresponding variants mapped to the Human genome (assembly hg19). You can again get these files and see where the neanderthal/denisovan variants lie in the haplotype file using the same approach. Finally you can then use these two overlaps to find where your variants had mapped and if any ancient variants had mapped in the same region. You must also make consideration for Insertions and deletions within the genome, so the variants may not map at the same location but it will map nearby.
UPDATE

You can check out this link for retrieving allele frequencies from 1K genomes, for the overlapping variants. I never meant that you would use the bed files for finding allele frequencies. What I outlined is;

Use the haplotype population regions as a template and overlap your significant affy SNP variants on it.
Next, because the Neanderthal and Denisovan variants are mapped on the hg19 or hg18 assembly, you map back those variants to the same regions
Find the regions where an affy SNP and an ancient SNP exactly overlap

What I understand from your comment is, you have already done step one. And you want to know the allele frequencies for a particular ancient variant. You need to do step 2 and 3 before coming to allele frequencies.
Then you can look up the allele frequency of the variants which mapped exactly at a position which had an ancient variant in the 1K genome vcf files.