Rooting your phone is basically the method of gaining administrator access to the Android operating system. Doing this allows you to install applications that you wouldn't normally be able to install. It is not without it's risks, I am inclined to say, but there are many guides out there that will tell you exactly what to do.
You can update your phone via Sony[Ericsson's] PC Companion software. This can be downloaded from the Sony[Ericsson] website.
Lastly, if you go to the Market - which is an app on your phone - and search for and install "Flash", this should sort your issue out. There may be an option in your browser that you will have to check to enable Flash content.