This is quite doable.
Just make sure of the following:
Any test data should be one of the shapes you have trained on.
Make sure you have balanced data. (You can introduce bias if required)
To make it more challenging you can create or find data like google street view images for numbers.
Segmentation and Classification, both are possible.
VGG should do ...
Using a relu function at the n-1 layer could be too constraining if you want your network to produce both positive and negative values. I am not sure about your image preprocessing, but I would first give a try to change (at least) the last activation function relu to leaky relu or tanh (an activation function that produce both positive and negative values).
Basically in machine learning, you can't take a dataset and want to "find any info". You have to specify which kind of info you want. In your case, you could do a complete model trying to predict, using the logo as entry, if the company is recent or old for example. Here you defined your problem. Then you'll need training data, so you'll either ...
Very likely no!
Machine learning algorithms aren't magic, they cannot see or find stuff that is not there.
We know for a fact that some trends and hints exist that link a companies exterior communication to it's industry e.g. social media companies like blue logos (think Twitter, Facebook, linkedin, etc.).
However for the most part logos, brand names, etc. ...