What kind of sostenuto pedal do you have?
There are two that I am aware of.
1. The actual sostenuto pedal where stepping on it after playing notes will hold those notes' dampers up thereby allowing those strings to vibrate. However, if stepped on and then notes are played, nothing will happen - the dampers will silence the strings just as normal. This action will hold any notes with dampers, not just the bass strings.
2. The cheap alternative to real sostenuto action: the bass string dampers are raised when you step on the pedal, and functions just like the damper pedal.
Depending on which of these two you have, you may have to disregard the sostenuto instruction because it will be impossible to follow if you do not have the action in 1. I have the action in 2, by the way.
Any advice is appreciated. Also, when using this pedal, do you use the other pedal, releasing the same as you normally would.
Yes, you can if you have the action described in 1. Since whatever notes were held by the sostenuto pedal, the use of the damper will raise all of the dampers but the sostenuto will keep those notes raised until you release the pedal.
You can also use it with the action described in 2 but the problem is with the bass string dampers raised, the strings will vibrate as the notes in the treble are played. If you don't know how this works, try this: Press C1 without striking the string and keep it depressed. Then play C4 and release. Notice how the C1 string vibrates after C4 was struck? Try it again but this time, instead of playing C4, play D4 while still holding C1. Notice that C1 doesn't vibrate when D4 is struck? This is what happens to all of the strings if you have the action described in 2.
Do they ever put BOTH pedal markings, or is it "assumed" that you would normally just know to use it as well?
It depends. From the music I've seen, whenever it says to use the sostenuto, that means explicitly to use the sostenuto but any other phrases played after the sostenuto can be played without damper or with. You are going to have to experiment with which sounds best.
The first severl measures are just eighth notes for the bass, so I'm confused at what the composer is trying to accomplish here.
If you mean that the bass note is played over and over, repeating the same note, then that just means he wants the note to be played without being silenced. The same effect can be had if you play the note without letting up so much that the damper silences the string before the next note is played. It sounds the same as using the sostenuto but is much easier to use the sostenuto to hold that note.