HI. Welcome to HCM
From your description and the location of the rashes, you seem to have seborrheic
dermatitis, affecting the oily areas of your body, similar to
dandruff on the scalp. Even if you do not generally have dandruff, oiliness of scalp and hair could have caused the rashes on back and chest. Change in climate, sweating, certain hair products etc, can aggravate the problem.
Dry skin and rashes on the arms, elbows etc can indicate an eczema or allergic nature, often seen along with seborrheic dermatitis.
For your hair try using an antidandruff
ketoconazole shampoo twice a week and a similar soap for the body which will reduce the problem. Do not apply oil, gels etc for long periods of time.
As for rashes on your arms and severe rashes on your body use a steroid cream lightly,like
mometasone for 7-10 days. Always keep your skin moisturized immediately after bath or shower with a good lotion like
cetaphil, or nivea. Your skin type is prone to dryness and allergies, so keep it free from harsh chemicals, fragrances. Use mild
glycerin or syndet based soap bar for bath over these areas.
Start with these treatment for a week or so. If you notice no change, or increase in rashes be sure to
consult a dermatologist and get a confirmed diagnosis. they can also rule out fungal infections, or do tests required to check for specific allergies, etc.
Hope this helps.