Acne does not make one more susceptible to skin cancer. Blood-red spots about the size of a pinhead have appeared within 2 weeks on various places on my body. It is more likely a sign that the blood vessels in the skin have become inflamed or there is a blood clotting problem. This can have various causes such as new medications, infections or other diseases. If the spots truly have appeared within two weeks this would certainly be worth a trip to the doctor because a visual exam by your doctor can distinguish most of these types of problems.
Most people have a few bright red spots on their body that are slow to form.
They form over years. These are non-concerning cherry angiomas. I recommend staying out of sun-tanning beds. The beds are not as harmful as they used to be. They are mainly UVA wavelength of light now rather than UVB, which is more harmful.