Province and assembly are common nouns unless they are specifically used as the name of something. Example: Henan Province, US Senate, the French National Assembly. I believe Senate is always capitalized because it refers to the specific Senate of either the state or federal governments. When used in a general way, province and assembly are common nouns. Example: Henan is a province in China. Our organization's assembly meets next week.

They're common nouns unless used with a proper noun, in which case they must both be capitalized *OR* when the word senate refers to an official governmental body. Senator Johnson represents his constituents in the US Senate. The word "senate" is a noun. The US Senate is currently on recess.

Province, senate and assembly are common nouns. Unless they start a sentence then they should be written in lower case. If you are talking about a specific one of things where it is part of the name, then it is part of a proper noun and should be capitalised. For example the Australian Senate (aka "The Senate"), should be capitalised.