Why isn’t a sincere heart enough? Why do outward deeds matter?

Because for every internal action there has to be an external action, and the value in that external action is achieved by the internal aspect. There is no sincerity without action, and no action becomes worship without sincerity.

Do good deeds count if a person has no faith?

In the hereafter, no, faith is the foundation that gives deeds their worth. But once a person accepts Islam, even the good he did beforehand is credited to him.

Why is purifying the heart (Tazkiyah) so important?

The Qur'an gives the glad tidings of success to the one who purifies himself, and names the gardens of perpetual residence as the reward of one who purifies himself. Some may think purification is just an option, but Allah commands the believers to rid the inner self of all evils and purify it. Just as there are forbidden acts, there are forbidden inner states like jealousy, greed, pride and hypocrisy.

Were the great Islamic scholars actually Sufis?

Many of the Islamic scholars to whom Allah gave success in their work were Sufis. To discard the works of those trained by Sufis would mean discarding “75 percent or more of the books of Islam.”