You deforest an area or a country or something - not a forest. You fell a tree, or all the trees in a forest; 'fell a forest' sounds to me a bit odd. 'Logging' is an activity that I wouldn't expect to see referred to with the verb 'log' and an object. So all your examples sound a bit odd to me. You could say 'The area will be deforested. They're felling all the trees. Come next Spring there'll be no work here, apart from logging.'
But 'deforest[ation etc]' is a rather formal word - used in geography texts and newspaper articles. I think in speech people would be more likely to say something like 'they're cutting down all the trees in the forest'; or, if the trees are small enough to be uprooted, 'they're uprooting/grubbing up all the trees in the forest'.
Br English warning: my views of general usage are rather parochial. Am English favours long words like 'hospitalization' (where Br E would use 'going to/staying in/being put in hospital). I wouldn't be surprised if 'deforestation' is in much more colloquial use in N. America.