Thank you for your query,
The cough is just looking like due to throat irritation
may be an allergic one as there is no fever or some time due to upper respiratory tract
infection which initially starts with a cough due to infection focus in the throat.
The rash on leg with hot due to evolving pustule or a skin infection
and may need topical antibiotic
cream application. But examination is important to say whether it is a pustule or a just papule.
So consult your Pediatrcian for further management.
Hope I have answered your query.