Do Shia Muslims have an obligation to wear a skull cap while offering Salat or reciting the Quran, as some people ridicule those who do not wear one?
No. There is no obligation to wear any cap while performing SALAH or reciting Quran. Women should obviously wear Hijab but there is no obligation on men to wear a cap or so. It is recommended for men who can to wear turban to do so while praying SALAH. No obligation at all to cover the head or wear a skull cap.
Wassalam.