Let’s first clarify some terminology:
A letter is a basic (often atomic) element of a writing system.
Which letter you use in a word depends on the orthography you use, but it does not depend on the typeface, font, handwriting style or similar.
For example in standard English orthography, the word fine always consists of the letters f, i, n, and e – irrespective of whether you use an ﬁ ligature or not.
A character is also a basic element of a writing system, but the term is wider than letter as it also captures numbers, punctuation, etc.
Unicode (and other text encoding systems) use characters as basic units.
A glyph is the basic (usually atomic) unit of a font.
When a text is rendered in a font, it will be composed of these glyphs.
Usually, glyphs are shapes that cannot be divided into simpler shapes.
In movable type, a glyph would correspond to a single movable element, also called sort.
For example, here are a few glyphs of the font in which this text is rendered: a, A, :, $, 7, ß.
A font may contain several glyphs for one letter, for example some fonts feature, both, a monocular and binocular g, and the designer can decide which one to use.
A ligature is a glyph that spans two or more letters.
This is usually used to connect the letters in some way to make them more pleasing to read.
For example, the ﬁ ligature contained in many typefaces is a glyph: you cannot split it since the f and i are connected.
It is debatable whether the term ligature can be applied to handwriting at all.
(Note that while Unicode contains some ligature codepoints due to historic reasons, ligatures are usually not considered characters.)
A digraph is a pair of letters that have a special meaning in some orthography.
For example, sh is a digraph in the standard English orthography, representing a different sound than just the combination of an s and an h sound (except for words like mishap).
Now to your questions:
I also understand that letters on their own like D or S are considered nouns or words.
This is not exactly correct.
There are words we use to refer to letters when talking about them.
For example the word zet (usually spelt Z) refers to the letter Z, but they are different things; you cannot speak a letter as such.
The difference between the two is the same as between water (the chemical substance) and the word water.
When connecting two letters or graphemes with a ligature as one glyph, is the glyph considered one word as D and S are, or two?
When they are joined, are these ligature glyphs named as D and S are, so as “dee” and “ess”?
For everything that humans want to talk about, terms exists.
Sometimes these are words; sometimes these are more complicated.
For example, in the above I used the term ﬁ ligature to refer to something and I suppose you understood it.
I leave it up to you whether you consider this a word or two words, as it doesn’t really matter.
You usually will not find these terms in the dictionary, as they are not relevant enough and people have little doubt about their meaning, spelling, etc.
When are ligatured letters considered their own characters, and when is a ligatured glyph considered two letters even though it is called one “glyph”?
As mentioned above, using a ligature does not change the letters per se.
However, it sometimes happens that an orthography features a digraph, and then a ligature for this digraph becomes so established that it turns into a letter.
For example, roughly speaking, the letter w originated from a uu digraph (hence its English name) and its graphical shape originates from the corresponding ligature.
In another example, the German letter ß originates from an ſz digraph and ligature; it is the final stages of a transition that took centuries.