Well, the spots definitely look like some type of bug bite looking at the picture. It does not have to be bed bugs, there are lots of different types of bugs and very small flying insects.
Contact dermatitis and eczema tends to appear as very dry, broken skin with or without inflammation on the skin. Spots sometimes can appear with eczema, but are are much smaller and will be in a group on the dry and irritated skin which can also break and bleed This would also be extremely itchy.
Do you have any pet animals in the house? Animal fleas can bite humans.
Flea eggs can lay dormant in carpets and soft furnishings for a very long time
As the spots you have appear only on your arms and hands, then you can rule out illnesses like measles. You would also feel very unwell and have a fever and spots on your body.
The antihistamines will work for you regardless of what causes these spots, because it reduces the histamine that is produced by the body.
I would suggest that you put all your bedding to wash on the hottest setting you can. Wash your pillow/s too. Wash any soft toys that you have in your room and if these are not able to be washed, pop them into a freezer bag and then into the freezer for a day or so.
Vacuum the whole bedroom, the curtains, skirting boards and any nooks and crannies, your whole bed and the headboard. Remove books that are in your bedroom.
If you do have a pet animal, ensure that you use the appropriate anti-flea stuff and also de-flea any places where your pet has been.
I am not saying that these are flea bites, but it is still worth taking all these measures.
With you mentioning that you have red spots under the skin, it may be worth considering whether these spots are caused by scabies. You mention schoolwork, so I presume that you go to school and it is very easy and common to get scabies from touching someone who has it. Sometimes with scabies you may notice little white tram lines. If it is scabies, you will need a special lotion to apply to the skin to get rid of them.
I would suggest you go back to your doctor or a different one at your surgery that may be able to identify what is causing your problem so that you can get the appropriate treatment.
Let us know how you get on.
You may have eczema or contact dermatitis. I would suggest having your physician refer you to an allergist and a dermatologist to determine what the problem is exactly. In the interim, there is over the counter hydrocortisone creams that you can use, as well as over the counter allergy medications, like Benadryl. I would not take either of them for about one week prior to the allergy and dermatology consults, so that they can return and the docs can see what they look like. Good luck to you!
Also, i've been getting these hard bumps on my fingers that are rock hard and hurt a little when i put pressure on them. the doc I went to didnt pay attention to them but they also went away when i took the antihistamines.