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First off, welcome to the forum jonv. Setup your tablet with the same Gmail account that you have on your phone, then go to Google Play then My Apps and swipe left or right until you see all and all of your purchased apps should be listed unless they are not compatable with your Nexus 7.
I think apps only auto-install on the same device, not sure if it's the device ID or the devices fingerprint that is used but none of my phone apps were installed on my N7 when I activated my N7.Honestly, I wouldn't of wanted them to auto-install. I do have some of the same apps on both devices but some either don't work on the N7, don't work properly or I just plain wouldn't want them on my N7...Oh, welcome to the Nexus 7 Forums jonv.
I've found it pretty hit and miss as to whether apps auto-install after flashing a ROM. I just flashed AOKP JB on my S2 and my apps didn't auto-install from GP but they did last week when I flashed CM10.Usually I cancel them anyway and restore with Titanium, it's much quicker to do a batch restore when flashing between ROM's than waiting for GP to install them especially if there a bad data signal.As for apps on your phone being the same as those for a tablet, it's probably best to install them through GP because any apps that are compatible with the N7 will say on the application information screen, "This application is not compatible with your device".Some apps do have second versions for tabs but most are multi compatible, the only app I can think of off the top of my head that has two versions is Swift key but the phone version does actually work despite a warning once you open the app. I'm sure there are others but nothing I use that I remember.
Sling media. If their customer support wasn't so bad then maybe they wouldn't get such a hard time for the $ of their apps and incompatibility issues.