This is not a trivial topic in the least.
The answer i could best give would address two parts of your question separately.
1) Blender and other programs may have different matrix setups and even animation and deformation matrix orders.
2) Deformation and orientation matrices are used to make models move as you are describing.
As far as i know monogames ability to load a model and gleem information is limited.
There are a few people who have posted code on model loading here in the topics.
I suggest you search for topics on model loading on the forums and take a look at others code.
I even took a shot at it i have yet to revisit my class files and clean them up i used assimp and found it however does have some problems so does blender which are too lengthy to discuss for me here.
My incomplete work can be found here,
It is most definitely a personal rough draft and not meant as a tutorial.
This does work if the model will load but has quirks depending on how the model was made which i haven't figured out a way to detect and smooth out yet. It requires you use the openassimp nuget package easily added from the visual studio nuget package manager. If you look in the code you will find the RiggedModel class which does the work and the shader RiggedModelEffect.fx in the content folder the other class file holds all the loading code and classes.
Loading the model is half the battle but manipulating it isn't really trivial either.
Here is another post within the post is a link to nkasts aether model loader which is if you can get it working im sure in a more complete state. As well as where i was at when i last worked on mine.