The four main schools of jurisprudence allow tawassul in all the 4 types.
- By mentioning the beautiful names of Allah
- One’s own good deeds
- Through a pious living person
- Deceased pious person
The first time tawassul through a deceased pious person (including Prophets and Messengers) was deemed impermissible was Imam ibn Taymiyya.
Yes, Muslims can pray for non Muslims. The two exceptions are 1) praying for salaamati, i.e., giving or saying the Islamic greeting of As salaamu alaykum and 2) praying for maghfirat after their death. The best dua for them is guidance and tawfeeq. We can also pray for relief from any worldly tribulations they might be suffering from.