Glad to have you around, Bob.
Baseball is a different game - different culture, so umps need a different approach at times. And many seem to use that same approach when they talk to each other on a board. Talk to one over a beer, and you'll get more of an understanding. So much for what I think things look like.
Anyway, you had some questions.
1-"My first question is about mechanics. R1 on first, stealing second. The pitch gets away from the catcher, either a WP or a PB (doesn't matter). Who takes the runner to third in this situation? Do I, as a BU, chase the runner to third? If so, do I stay outside, or do I come inside? Last year, I found myself chasing runners from the outside many times, and even though I am able to hustle enough to do that, it still didn't just feel quite right."
For Fed, the responsibility (with no other runners on) of R1 at 3B on this play belongs to the plate ump. In an organization where the responsibility is the base ump's, the plate ump must be ready to help, 'cuz BU can't always get where he needs to be to make a valid call. But in that organization, the plate ump is helping & the responsibility is the base ump's. In Fed, it belongs to the plate ump.
When I am the base ump, I will stay outside. If I go inside, I will be too far behind R1 and will not be able to get the angle I want. But, I am able to outrun the vast majority of the players I see.
2-"Second questions concerns walking the line as a BU. I gather that this used to be a mechanic for ASA, as my instructors at a recent ASA school kept referring to it as a return to a previous mechanic. It was a little uncomfortable for me, but I'm sure I just need to get used to it. Did Fed change to this mechanic also?"
This can be a very good mechanic, if you take just a single step. I want my eyes still as the pitch is released - so the camera I'm picuring the play with is still. Start on the line, with your right foot touching/near the outside of the line. As the pitch starts, take a single step forward with your left foot. Your left foot is your pivot foot and doing this buys you a step advantage. I'll do this in all sanctioning bodies.
3-"...What exactly is a blocked hat? I kinda think that I know what that means, but would appreciate someone enlightening me."
A blocked hat is one with a crease in it at the brim. Some of us think that it looks better. Here's a link to another site that many of us visit & participate in. There is a writeup on how to fold & block a hat.
http://eteamz.active.com/softballumpires/