Is there a client side service required?

No, there aren't any client side services that need to be running in order to remotely deploy and install your packages.

Is there a limit on the number of machines I can deploy to?

No, your license applies to the application for creating packages and scripts. You can deploy packages to as many machines as you like.

What other requirements are there for running the software?

SQL server software is required in order to run the software, since it stores information in a database.

This can be the Free SQLEXPRESS software offered by Microsoft.

Simply install SqlExpress and attach the neatpackage.mdf database file to it using the downloadable installer.

Do deployments run in the background?

They can be run both interactively, and non-interactive.

If you prefer, all deployments and installations can be hidden from the client users on their machines and run "non-interactively" in the background. This allows the users to continue to be productive while software is being pushed to their machine, but requires that all installation pieces be run in "Silent" mode, or with response files.