Actually BEAD usually has slightly less tension that EADG. Much of the tension in EADG is from the tighter D and G strings, The B string is tuned lower and tension is not as great. As you can see here:
Tension for a D’Addario standard .045-.130 set highlighted there, tension on the G string is significantly more than the B string. So much so that you can even go up to a 0.145 B string and have less tension than EADG.
Correct.
I filed my nut with a cheap round diamond file I had lying around. Worked perfectly. You just need to be careful and go slow. Maybe buy a couple nut blanks?
Unless you plan on doing this to many, many basses, I wouldn’t waste the money on the nut files. YMMV.
You may need to widen or replace the bridge saddle with a wider one, but probably not. I didn’t.
The only negative is the lack of the G string. There’s a whole ton of positives around that B string. The bass loses five semitones in the top end and gains five in the low end. Whether or not you like it is up to you 
You will need to relearn (or re-remember) the fretboard like it’s the top four of a 5-er.