How do I swap between branches
If you're used to GIT than the 'heavy weight' branching in TFVC can be a bit confusing. There is no real "Switching between branches" as you've encountered. You map a branch to a local folder and by opening the files there you're "working on that branch".
As Lee points out you can create separate workspaces for each branch, which will isolate the work areas for each. If you're using a Local Workspace, each workspace gets its own "/tf$" folder, the TFVC equivalent of the "/.git" folder.
There's a couple of documents on MSDN that explain this in a little more detail:
How do I check in
A changeset in TFVC is the equivalent of a commit in Git, it's a logical set of changed files that is committed/pushed as a whole, or not at all. But just as in Git, you can commit all the changes to your local work area at once, or you can exclude certain changes from the first commit and stick those in a second.
In TFVC you'd normally try to commit a logical set of files that fixed the bug, achieved some goal etc. Though it's still possible to check-out/check-in files individually, chances are much higher that you'll actually cause the sources in the main repository to be in an inconsistent state that way.
See
As for your second question
Depending on how far you'd want to go, you could setup Team Build to actually build the application and to take the configuration from a specific location during the build process. That way you wouldn't have to store the configuration for your production environment with the development settings. Configuration files can contain sensitive information, you might not want to have them in Source Control, except for the development versions.
You can also store the config files in a special folder in each branch and make sure that each time you merge them, they're updated accordingly.
And you can, as Lee mentions, look into Config Transaformations. which apply some XSLT to your config file in the build process. That way you can have multiple config files stored in each branch and the selection of your "Configuration" in Visual Studio will define what the final config looks like.
See: