Do I ever install the Feature in my SharePoint project manually, or is that handled when I install the solution? Am I only supposed to have one overall, or one per webpart?
There is nothing call Install Feature. When you deploy a new solution this is what happens. It first add the solution to farm. Then deploy the solution. If you have set the features to activate on the deployment in the Feature settings, it activates.
You do not need to have one feature per webpart. It depends on the requirement. You can even handle with one feature for many webparts. If you need 3 webparts for a one feature and 5 in another, even that is possible.
You should structure your features in a way that makes sense to your project. The key aspect of features is that they can be turned on and off on a site. The simplest thing to do is to put all your stuff in one feature, but that may or may not be the best solution in your case.
Whether a feature is automatically activated on deployment depends on a few factors. If you are deploying through Visual Studio, there is a setting on the project that determines what happens on deployment. The default is to activate the features. You can see the settings by right clicking the project -> properties -> SharePoint tab.
If you are doing a "real-world" deployment via PowerShell, you'll want to activate the features on the sites where you want them activate.