No and no. Forest preserves only provide their benefits if they're in the BFC (big fat cross), that is, the tiles the citizens of that city can work--the first 2 culture rings, basically.
Forest preserves are very weak tiles; they provide a very negligible production return--IIRC, no improvement over the base tile's food and hammers--for a very small benefit. There are better ways to improve forest tiles (including chopping them), and better ways to obtain and increase happiness. The only time I bother with forest preserves are around the National Park city for the free specialists--and in some games I don't bother to build the NP.
In some non-ancient starts the NP+NE in Environmentalism is a very strong play. Environmentalism increased the commerce output of the FP's to make those tile worth using. Otherwise, NP is something I usually only bother with in OCC games. NP+Globe Theatre in a OCC is very nice if going for a long victory condition.