How does interacting with the DNA result in a cellular response, does the signal transduction result in the production of new enzymes to carry out the response?
Genes can be either activated or suppressed; I don't know how much you know as this is unit 4 stuff. So if a gene is activated, that means that mRNA is copied from it, and thus that protein production begins. If a gene is suppressed, mRNA copying stops, so protein production stops. So the hormone essentially controls production of a specific enzyme, and thus whether a specific reaction occurs or not.
So (this is hypothetical, obviously untrue example) let's say that we need more ATP produced in muscle cells because the person has started jogging. A specific enzyme is involved in one of the steps of ATP production. So a hormone might be sent to muscle cells, it goes to the DNA, activates a gene segment so the enzyme is produced, and therefore the enzyme can start working to produce ATP. When ATP isn't needed so much, another hormone can be sent to suppress the gene, stop this enzyme production, and thus stop ATP production.
Is that intracellular responses or intracellular enzymes which will carry out the response? or is it the same thing?
Yup, same thing.
Also why is hydrophilic hormones faster (in terms of response) than the lipophillic hormones? Is it because the lipophillic ones have to enter the DNA and code for a new enzyme to be produced, and then carry the response, whereas the protein based ones directly target intracellular enzymes in the cytosol, hence the response is much quicker in general?
My guess is (though I'm not sure) that hydrophilic hormones trigger a lot of second messengers - so with the lipophilic, one individual molecule travels through and works directly, whereas with hydrophilic lots and lots of other molecules are triggered to begin working, so the response is faster. Dunno.