If you are logged in on your Apple TV (2nd Gen) with your YouTube user account (such that you see your user ID in the YouTube menu), then you should be able to navigate the cursor to your user name (it's listed at the bottom, after "Most Viewed, Featured", and other menu options). Select your user name and hit the select button. You should now see your options as a logged-in YouTube user, including Favorites and Subscriptions. I can see Favorites on my ATV 2nd gen. . . not sure why you cannot.
As for the videos on your YouTube page, for some reason the Apple folks have not included the ability to watch your own YouTube videos on your Apple TV. I don't know why, and am annoyed about it, especially since the YouTube app for the iPhone (also made by Apple, not YouTube) has a "My Videos" section, while the Apple TV "app" does not. A workaround is to just mark all your own YouTube videos as favorites on the YouTube website, and then you'll be able to see them amongst your other favorites on your Apple TV. I don't think you can "subscribe" to your own video feed, so that won't work.
Frankly, I think the easiest way to use YouTube with the ATV is not to use the built-in YouTube functionality on the ATV itself, but instead use the YouTube app ON YOUR iPHONE. Do all your searching on that iPhone YouTube app, since it allows you to see "My Videos", as well as your favorites and subscriptions, and also makes it easier to search for other videos. Then, when you want to watch a particular video on your bigscreen, just use Airplay to send it to your Apple TV.
Otherwise, you can do what I do: If you see a video while browsing the web on your computer that you want to watch later on your TV, just favorite it without even watching it. Then you should be able to see it on your Apple TV. You can remove it as a favorite if you only want to watch it once.
The remote app for your iPhone is just called "Remote", and is a great way to control your Apple TV from your iPhone. It is made by Apple, and available free on the iTunes App store, accessible directly from your iphone. I prefer a physical remote (either the one that came with the ATV, or a universal remote which your Apple TV can "learn") to move the ATV cursor around and control playback. However, the one thing that the iPhone Remote app is best at is entering text because you can use the iphone's onscreen keyboard instead of having to painstakingly enter text one letter at a time with the ATV's onscreen keyboard. Basically, I use my universal remote to navigate around my ATV for most things, and then if I ever want to search, I just run the Remote app to enter text.
