Culture flips happen based on the culture of the province owner. If the owner of the province has high Stewardship they happen faster. If a province was conquered in holy war that also speeds up cultural conversion.
Religion flips are not automatic. They happen by having a Court Chaplain Proselytize in the province. AI characters will also use their own Court Chaplain to Proselytize in any provinces they own (starting with the capital) as well as in the provinces of their vassals (once they've fully converted their personal demesne).
However, single province counts may not have a Court Chaplain. The AI seems to prefer giving Court Chaplain position only to temple vassals and close family members, but will use them for other jobs first if it doesn't have enough other vassals. So if you're creating a new count, he'll generally need to have 4 other vassals (of any type) in addition to a temple vassal in order to reliably have a Court Chaplain. If you set up a new duke, he'll be able to call both on the baron tier vassals of his provinces and his vassal counts for jobs, so they usually do have a complete council.