Yeah, it can be sort of like "and, you know." It's a lot closer to が, けど, けれど, けれども, のに, etc. than it is any of the ones you listed (except it has an additive meaning ("and"), not a contrastive one ("but/although"), as I said above).
In terms of construction you just add it to the end of a sentence like any of the ones listed above (not necessarily just verbs).
A: 先生!
B: いや、彼は学生だけど。
彼女は優しいし、きれいだし、頭もいい。
明日のパーティ行く?俺も行くし・・・ (alright, someone check that one for naturalness)