Disclaimer: this is for Civ 6 Vanilla on PS4
Removing woods affects only production, not food. Also, until you get the Conservation civic and can make National Forests the woods do not add an appeal to the tile they are on, only the adjoining tiles. Once they are considered "Old Growth" they gain one appeal on their own tile, but not until the Modern Era when you gain Conservation. New growth only ever adds appeal to adjoining tiles, never its own.
No, a Lumber Mill does not affect if woods are considered old or new growth.
There are some errors in the original post and omissions of information that were not addressed by anyone yet... so for clarity's sake:
The tile you are talking about is a grassland with a hill and woods. The grass (2) vs plains/tundra (1) vs desert/snow (0) affects the food, not the production. The hill gives 1 production and allows a mine, the woods give an additional production and allow the lumber mill, but block the mine.
Also, harvesting the woods (anything really) is not a flat 50 production, but is a base of 25 (iirc) and then scales with research completed based on game speed. I am going to guess you are late ancient/early medieval. The game I am playing right now is in early Atomic on standard speed and I get in the 140's for harvesting old growth woods. That harvest is again boosted by any +production % policy cards you have on and apply to what is being produced. (30% Builder card will net you an extra 30 production per 100 harvest value if the city that owns the tile is making a builder. Same goes for the 15% wonder policy. Make sure you get the most out of those woods =D)
Unless you are planning on making that exact tile part of a National Park you should almost always cut down woods on a hill. There are few reasons to pass up on the instant gains + per turn benefits of a mine on a hill in early/mid game. However, if that tile is next to a river the lumber mill will give equal production value to a mine (+2 from mill with a river and you keep the 1 from the woods) it also won't decrease the appeal of surrounding tiles, and you save the woods harvest for a later time when it is worth a lot more than 50 or you have a better policy in place to take better advantage. Also, if next to a river or you have Steel researched it only costs 1 builder charge to place a mill, vs 2 to harvest then mine, or three to harvest, grow forest, then mill.
After you get the Conservation civic (early modern era) you can make new growth forests that give zero production for harvesting, but add 1 production and allow the lumber mill. Once you get this you should cut down all your old growth woods that are not part of a park and replace with new growth and pop a mill on them. Do this while the builder policy is in place and it is snowballing profit (one charge gets you a whole new builder) Once you get Conservation, the Steel tech is the same exact era on the science tree and gives +1 production to lumber mills, making them always as good as a mine and better than a mine if next to a river (excluding adjacency bonuses to the industrial zone).
There are a few pantheons and religious beliefs that involve woods and rainforests, but are pretty weak so don't pick them in the first place. Goddess of the Harvest gives faith equal to the harvest value of anything you harvest including woods, rainforests, marshes, and bonus resources like wheat and rice, and is a very solid choice. There is also the Goddess of the Earth (i think) that gives 1 faith per point of appeal, but the tile has to be worked. There are a few wonders that involve woods/rainforests and I already mentioned the national park which receives tourism based on the sum of its 4 tiles' appeal value, and old growth woods add 1 to the appeal of the tile it is on and all adjacent tiles whereas the new growth only gives an adjacency bonus, nothing for the tile it is on.
I am about to switch all of my mines on hills next to rivers to lumber mills cuz I just got conservation... which is why I was googling the topic mines vs lumber mills.
Also... Unrelated to the benefits to the tile itself... any environmental minded civs don't like it when you harvest woods... But so? I usually kill them bevause they always denounce me amd sometimes attack.