Q: Is there any way to watch YouTube (or a similar website) directly on my PSP with the web browser?
A: No. Sony hinted that the functionality would arrive in a later firmware upgrade, but that hasn't happened yet. You have two options with YouTube: you can convert the videos and save them to Memory Stick, or you can use Remote Play. If it works for you, the Remote Play internet browser is able to stream Flash video to your system.
Q: Are there any free programs which can change Real media files to be played on my PSP?
A: People still use RealPlayer? The format, to my knowledge, is pretty much obsolete. You can use Media-Convert to try to use them, but remember: you can't convert DRM-protected files, nor can you convert files that have no video data in them. If your .rm file looks really tiny (just a few bytes), it won't convert.
Q: Do you have any suggestions for any good games that utilize my router?
A: The big one to get is probably Metal Gear Solid: Portable Ops. Other candidates include SOCOM, Medal of Honor, Ratchet & Clank and Killzone: Liberation.