I'm pretty sure by factory resetting all your apps are lost. What you can do is login your account in Play Store and click My Apps, and redownload all your lost stuff again.
I'm not sure though if plugging in an SD card will solve your issue with downloading apps, but when my phone with ICS didn't have an SD card even though it has an internal storage, it wouldn't let me capture photos. My bet is, yes you need to plugin an SD card. Try borrowing one first and see if that solves your problem, then buy one. I think you should definitely buy one though, for your media like music, photos and videos.
One more thing, if you don't want to lose apps, I suggest you download Appmonster. It saves your applications as apk files in your external storage. What does this mean? You can backup all your apps, revert to older versions of applications, and even save app memory. Just uninstall the apps you don't need and install them whenever you need them. All this, without having to go to the Play Store.