Since mobile development is agile, this needs to use test automation.
Mobile development can be agile. It is not a given. Test automation can be done in any paradigm.
Do emulators work 100% accurate?
I can't say 100% but they are getting more accurate (I haven't had an issue since iOS 6). For Android, make sure to use an AVD with Google Play API services.
I have experience in Selenium webdriver/grid and c# scripting
Then you'll be at home writing Appium/Java tests. To me, C# and Java are basically the same. Especially when using common libraries like Selenium. Alternatively, I use Xamarin to test iOS and Android with great success.
I am not sure about how cross-device/platform work in mobile testing.
I keep my desktop browser and mobile tests separate. They are different products and behave differently. I also keep my iOS and Android test projects separate.
Any tips on mobile testing and automation in general?
Testing on mobile isn't different. As an example: an input is still an input and can be tested the same.
One thing to remember is that automated testing on mobile is like Selenium/browsers in that it just controls the app (or browser) and not the OS. It can be a chore to do things outside the app like disable wifi or handle system alerts.