I'll address your question about logistic regression first because it's more theoretical:
When you perform logistic regression, you're computing some value: y = a + bX where a is a bias term, b is a vector of weights, and X is a feature vector. You then can compute a probability by applying:
P = (e^y)/(1 + e^y)
This P is the output you see, between 0 and 1.
In a hand-wavy sort of way, you can think of the b values from our formula for y above as the "marginal effect" of that particular feature. In plain English: b_1 is the amount that a unit change in x_1 will affect y.
This means, that these b values are scaled by the magnitude of the X feature they're associated with.
Lets think about this in terms of economics. If I want to model Consumption: C as a function of Income: Y, then I would have something like: C = a + bY.
The value b tells me how much a single dollar increase in income will increase my consumption. If my income is currently $10 million, a single dollar increase in income probably wouldn't mean I spent an extra dollar in consumption. However, if my income is currently $1000, then my consumption is more likely to increase by nearly that full dollar.
The reason I'm giving this example is because the weights that are computed in your logistic regression are inherently taking the scale of each feature into account during training. If you normalize the predictor variables, then these weights will be scaled up (or down, depending on the original values of the predictors) accordingly, and you'll get the same output.
That being said, if you have some extremely high variance in the values of your predictor variables, your model may be worse at making predictions for those values away from the mean, and this is something you'll certainly need to diagnose, however normalization won't help you out here.
Let me know if you need me to clarify anything above, or if I skipped something. I'll look into your question about the Random Forest model's output in the meantime,
Cheers!
