memeta

Nup.   pGEX vector contains the lacZ gene, so will produce blue colonies even upon desired insertation.

First, I would recommend always writing primers in the 5' to 3' direction.  Suppliers expect that order, and it is also very conventional in papers.  If you rewrite your reverse primer in that orientation, it is:
5' TCG GAATTC   TCA GTC TGA GTC AGG CC 3'
The binding portion of this primer seems a bit short (17 bp), but given the GGCC at the 3' end, I would probably leave it, especially since the next base would be C.  If you knew bases 3' of the end of the sequence you have given, I would add 2-3 just before the TCA in the primer above.

The forward primer is wrong.  The restriction sites are always at the 5' end of the primer.  You want something like this:
5' TGA GGATCC   ATG GAT GAT TTG ATG CTG 3'
Here, your short binding region really would be a problem.  I would recommend extending the primer by adding the next few bases:
5' TGA GGATCC   ATG GAT GAT TTG ATG CTG TCC 3'

Are you sure  you want to use BamHI as an enzyme?  It can't be heat killed, unlike most of the alternatives.

They are absolutely correct, however, the first open reading frame should be the one that is transcribed the most; the liklihood of skipping decreases as you get further from the promoter.  If you are using a short tag (e.g. V5) or have particular sequences in your gene that the promoter uses to enhance transcription, this could be a problem.

Find out if your GST tag has a kozak sequence - this should (in theory) minimise any chance of skipping.

It depends on if you are tagging N-terminally or C-terminally, or both.  In the case of an N-terminal tag, the GST will already have a start codon, but you can still leave the one found on your gene in, if you like, it will make a methionine in the aa sequence.  You can also cut it out if you like, it probably doesn't make much difference.  Many people would say to remove it to prevent skipping of the RNA polymerase.  You should include a stop codon for either of these situations. case!

For a C-terminal tag, you will need to include a start codon to make sure that the gene transcribes.  However, you must remove the stop codon to prevent the transcript terminating before the tag.