Can the Imam offer the Salat (prayer) with only those who are present (for the prayer)? And can be deliver a Khutba (religious talk) on Friday if it is raining?

Ibn `Abbas addressed us on a (rainy and) muddy day and when the Mu'adh-dhin said, "Come for the prayer" Ibn `Abbas ordered him to say, "Pray in your homes." The people began to look at one another with surprise as if they did not like it. Ibn `Abbas said, "It seems that you thought ill of it but no doubt it was done by one who was better than I (i.e. the Prophet). It (the prayer) is a strict order and I disliked to bring you out." Ibn `Abbas narrated the same as above but he said, "I did not like you to make you sinful (in refraining from coming to the mosque) and to come (to the mosque) covered with mud up to the knees."

Can the Imam offer the Salat (prayer) with only those who are present (for the prayer)? And can be deliver a Khutba (religious talk) on Friday if it is raining?

Narrated Anas bin Seereen:

I heard Anas saying, "A man from Ansar said to the Prophet, 'I cannot pray with you (in congregation).' He was a very fat man and he prepared a meal for the Prophet (ﷺ) and invited him to his house. He spread out a mat for the Prophet, and washed one of its sides with water, and the Prophet (ﷺ) prayed two rak`at on it." A man from the family of Al-Jaruid [??] asked, "Did the Prophet (ﷺ) used to pray the Duha (forenoon) prayer?" Anas said, "I did not see him praying the Duha prayer except on that day."