Does sneaking slowly produce less sound than sneaking regularily

You can change your base speed using the console, which you can open using the ~ key (or whatever key is to the left of the number strip at the top of your keyboard).

Type player.setav speedmult x where x is the number you want for your walking speed; the default is 100.

You can either increase or decrease this number. Increasing it higher will result in super fast running which man ruin the reality of the game.

I'd recommend player.setav speedmult 125.

In my opinion the default running is way too fast, and the default walking is too slow. While holding Alt to sprint is perfect. The only other way I know to edit these values is to open the core files, like XML with a text editor, but I won't go into that much detail here. You should be able to Google it.

Yes, it does help. I have personal experience with it, but no links to 'official' words on it.

The basis is that I wear heavy armor and sneak only on occasion, and noticed once I switched to the Steed, I was able to keep the 'running sneak' active around enemies much longer. My sneak skill is low 20s, and had not increased before I noticed. I do not sneak much, so when I do on that character I really notice the difference (I was actually removing all armor to try and sneak on him before getting the Steed stone).

As a side note, remember that weight is not the only thing that relates to sneaking. I have no perks on this character before OR after steed to assist with sneaking. But there are several available, as well as spells/potions.