Is a new Muslim permitted to perform some of the prayers at home and some at the mosque to avoid aversion, and is there any sin upon me for that, and what are the appropriate educational steps for him?

Start with the obligatory duties (fara'id) and the obligation to abstain from prohibitions (muharramat). Progress gradually in teaching him. Teach him how to pray, then explain to him the virtue of performing it in the mosque. Focus on the merits and fruits of Islam. Encourage him to adhere to the Muslim community and the company of righteous people, and to draw closer to Allah through abundant worship. Be a role model for him, and treat him with kindness.