Does offloading apps and then reinstalling them keep the same version or does it update it?

My experience is that if you reinstall an offloaded app, the last version you had is what is installed. But you will immediately be offered an update to the current version if there had been updates in the meantime. You can choose not to install the update - assuming you don't have automatic updates enabled.

Yes - offloading differs from deletion only in that associated App data is not removed from the device when an App is offloaded.

Reinstalling an App will always install the most recent version supported by your device.