GEN file format, SNPs and alleles
From what I could find the .gen file format for SNPs was developed for the IMPUTE2 program and is generally meant to be a flexible file format for SNP data. The description of the file format is supposed to documented here (multiple pages pointing to that link), sadly that link is broken / the site is removed. 
However, I found some other descriptions of the format:


  In the GEN format each SNP is represented as a set of three
  probabilities which correspond to the allele pairs AA,AB,BB.


from here


  The genfile contains predictor values, one row per predictor.
  --gen-skip indicates how many header rows (typically 0 or 1);
  --gen-headers indicates how many header columns (typically 0 to 5).
  --gen-probs should be 0, 1, 2, 3 or 4:
   0 - haplotypes - predictor values should be "0 0", "0 1", "1 0" or "1 1"
   1 - dosages - predictors provide the (expected) number of A alleles
   2 - two probs - providing probability of being AA or AB
   3 - three probs - providing probabilities of being AA, AB or BB
   4 - four probs - providing probabilties of being AA, AB, BB or NA, where A and B are the A1 and A2 alleles


from here

Like finswimmer already wrote in his answer, it is very important to be clear about what is considered an allele in the context of SNPs. For the .gen file format this is the identity of the base at a given (SNP) position and since humans have a diploid genome, the allele of a specific individual can be one of three versions (usually called AA, AB &amp; BB).

Regarding your specific points:


  Since each SNP has associated allele A and allele B, I assume that the file only contains Causative SNPs, and not Linked SNPs, since the former reside in the gene, and the later don't. Since each SNP in the file has an associated gene (actually genotype, some combination of allele A and B). Correct?



The alleles in the .gen file refer to the respective SNPs not genes.
There is no reason to assume that an given .gen file only contains certain SNPs; if it does this is not due to the data format.  
The SNPs in a .gen file do not generally have to be linked to a gene.