I could definitely see the orcs turning the pine forests of Dorthonion into charcoal and carting it off. I don't think that's *why* they're in Dorthonion, but if they are there anyway for strategic military reasons, that's one activity they could be engaging in. Stripping the land of whatever resources it has would be part of the utilitarian way orcs (and Morgoth) would think about the situation. Likewise, if they want to do some mining while they're in the area, why not? But they didn't come there to mine.
In Morgoth's opinion, all of Arda belongs to him personally. Anything they find there is already his. He's already laid claim to EVERYTHING, so taking what is his would be something he could command his armies to do.
Coincidentally, I was recently reading some fanfiction which featured palearctic survival practices in the Altai by a single man travelling alone. Food matters, of course, but the focus on getting out of the cold was also very prominent. Pine-needle tea thus serves the dual purpose of providing some vitamins, and warming you up from the inside. Most survival guides are going to assume you aren't opposed to trapping and eating animals, though, so I agree that Beren's practices are going to deviate from the usual. He's in a recently-inhabited area, not a strict wilderness area, so that gives him more options. We aren't going to detail exactly how he survives, but showing him accessing some food sources while he's an outlaw should demonstrate to the audience how he is living. I am not sure how we're going to convey the vegetarianism, unless he refuses a meat dish in Doriath and/or Nargothrond. Not showing him killing and eating animals won't make an audience assume he isn't doing it off screen, and, what? are we going to show a rabbit hop up to his fire and him kindly tell it it has nothing to fear from him? He doesn't have anyone to talk to while he's a lone outlaw, so explaining what he's thinking on screen isn't going to be particularly easy.
But yes, orcs eat meat, and are in favor of wanton destruction, so leaving cut trees lying and dead animal carcasses left to rot is part of how you can tell they are living in an area. They are...wasteful and rather uncaring about cultivating resources.