Very sorry to hear that you are going through this. Your GP should listen to you and not put it down to mental health issues, that is very unprofessional. Itchiness can be a sign of low thyroid (can cause very dry skin), or a liver problem that can happen during pregnancy. Hormones go all mental, especially in early stages of pregnancy, and that could be causing dry skin.
If you're finding it hard to eat with the sickness see if you can manage soups (there are several with quite a bit of protein), try to drink lots and rest as much as possible. Sickness can contribute to fainting feeling, also blood pressure changes with pregnancy which makes a wonderful cocktail of feeling somewhat shit for some women!
I would talk to the practice manager at the GP surgery and see if you can see another Dr. Do you have a midwife yet? If so, talk to them about it - if not I think you may be able to self-refer to them.
I'd also suggest ringing your local early pregnancy unit (EPU) to explain the problem - they will be able to give advice and help with testing or anything. If they mention going to GP then tell them that you were brushed off by them.
Last thing I would say is to get yourself some Epiderm. It is a really mild emollient that is great for newborns and eczema-prone skin, so is fine for use in pregnancy. You can wash with it, use it as a cleanser, and as a cream, so is really useful.
It is very wrong that the Dr is not listening to your concerns. I hope you can get some help from midwife/another GP soon, and start to feel better.