a. Is Thompson's the brand to use, or are there "better" products
available
?

You get what you pay for, and Thompsons is cheap.
That said, I have a dock on a lake. Your deck probably is less exposed and
gets less traffic. I have treated it 3 times in the 6 years I have owned
it, using the cheapest stuff I could find. Once Thompsons, once something
from KMart, and once something from BJ's.
The first time it sucked up the solution like crazy; presumably the previous
owner hadn't treated it.
After that it really didn't need it, as water still beaded up showing the
coating was still effective, but I thought 2 year is long enough to go. I
had time, and the stuff is cheap.
So, while Thompsons is not the best, it lasts more than 2 years on my dock.


b. How is this stuff applied ? Just swabbed on with a big, cheap,
paintbrush, or... ?

I use a roller on a broomstick. It is messy and wasteful, but the stuff is
cheap enough not to worry about. A brush is good for corners.
I have sprayed it, but the roller is easier and faster.

How about spraying it on ? If so, would you still need to smooth the
liquid with a brush ?

c. The underside of the Deck is high enough to walk under.

Should I also apply this stuff to the underside ? Brush or spray ?

The underside does not need it. It wouldn't be a bad idea if you have the
time, especially if it will get wet; but it just isn't worth the effort.

d. How does one get the sides of the planks, seeing that there is about
an 1/8 inch gap, only ?

Sides don't matter, as water will just flow off. Besides, there will be
enough slop from the roller to get the sides.
End grain is the most important though. Try to get all the end grain
several times.

Important to do the sides too ?

e. Does pressure treasted wood absorb any of this product, or is it
pretty much of a surface treatment ?

If you haven't treated recently, the PT will absorb it like a sponge. If you
have treated, it is pretty much a surface treatment.

Just out of curiosity, how about regular wood: surface only ?

You mean not PT? Sure, why not?