Thankfully! A new member that actually does use the search functions on the forum. Wish there were more like you.
That said, from the browser on the GN2, the only realistic way for you to be able to type in a search on the drop down is to go through Advanced search. Even though the S-Pen is great for being able to pull down drop downs on Full desktop versions of a site, it doesn't seem to have the ability of holding the drop down menu open, while inputting text (like a PC or mac).
Just in case you weren't aware of it, there's an app called Tapatalk that is connected to many forums on the internet. Many people (myself included) use that to post on forums, from their phone (rather than the browser).