Thank you for asking your query at DoctorSpring.
There are number of causes which can cause rashes and nausea/ vomiting in children. Most of them are infections - mostly viral infections. Viral infections can also cause pain and swelling in the joints.
Without further testing and examination it is difficult to corner towards a single diagnosis. The possibilities are Roseola, Erythema infectiosum, Measles, early Kawasaki disease etc. Please note these are just possibilities and your child do not have any typical sings or symptoms of these.
You can wait for 12-24 hours and see how this evolves. If the rash progresses and/or if there are warning sings like severe nausea and vomiting / fever / sever joint pains/ high coloured urine / frothy urine / diarrhoea / lethargy - you should get medical help.
Also if the child has not received vaccines / missed immunisation you should get medical help asap.
Hope this helps
Please feel free to ask followup questions.