A number of companies package SQL Express with their product, and run the SQL install during the product install. In most situations it is best to install a named instance of SQL, and try to pick a name that relates to your product and is unlikely to be used by a different product - this way you have an environment that is more likely to stay table.

The SQL Express install media does not include any pre-req software. Some people package a .Net3 install and run this if needed, but Windows 2012 and above need their own .Net install media and packaging everything needed for every version of Windows you want to support with your product can easily end up with over 1GB of pre-req software. You need to decide how you will deal with the pre-reqs.

You need to talk to Microsoft about licensing. As far as I know you can package SQL Express without any licensing issues, but you need to hear this direct from the vendor.