Is it ever appropriate to use a dotted quarter rest vs. an eighth note rest along with a quarter rest?
There are various opinions about this.
The "ultra-orthodox" view is that dotted rests are not even a thing. They don't exist. So you can't use them anywhere! The fact that people write music using them (and computer music notation software permits them) just demonstrates that many people don't know (or don't care) what they are doing. But you don't need to go around correcting their mistakes all the time, because some deity or other will probably have special punishments reserved for them in the afterlife :)
A slightly less extreme position is that dotted rests can only be used for complete beats (or multiple beats) in compound time signatures like 6/8 or 9/8.
That may seem extreme, but it does have some practical common sense to support it, because with hand copied music and/or in poor lighting conditions, dotted rests can be hard to read. Many music publishers followed that convention up to about 1900.
A more mainstream opinion today would be that dotted rests can be used if they are at the start of a beat, or a subdivision of a beat for short rests. For time signatures with an even number of beats the situation where a dotted rest starts on a beat in the first half of a bar but extends into the second half is not allowed.
So in 4/4 time you can write a bar of music as a dotted half rest + quarter note, but not a quarter note + dotted half rest. The second case should be quarter note, quarter rest, half rest. And similarly you can write a dotted 8th rest + 16th note (starting on a beat) but not a 16th note + dotted 8th rest.
That rule usually makes it clear where the beats are in complicated rhythms.
The most permissive opinion is that you can write whatever you like, of course - and that is true, but don't expect other people to read it without getting the rhythm wrong.
Incidentally, Glorfindel's second example is generally considered to be wrong, because when a beat is divided into 3 parts, the second and third subdivisions should not be combined into one rest. Again, the practical reason for this is to help show where the beats are in a compound time signature - a quarter rest will always be at the start of a beat. In the example, the visual appearance of "quarter rest + 8th rest" suggests a whole beat's rest, but it doesn't start on a beat.
We are currently in the middle of a progression from 'NO!' to 'same rules as for dotted notes'.
(There's actually a similar progression in thinking about notes. We all know the 'show the beat, show the half-bar' rules. But a more linear approach is becoming acceptable in some styles.)
In an examination situation, I suggest you play safe and reserve dotted rests for whole beats in compound time. In real life check which way the wind is blowing in your particular musical style!
Yes; rests aren't any different from notes here. There are situations where a dotted quarter is preferred; there are situations where a quarter + an eighth (or the other way around) is the more natural notation.
I don't have a concrete example from existing music, but consider the following example, a 6/8 time signature, where the rhythm is such that (most) measures are divided into two parts of 3 eighth notes. A dotted quarter rest is appropriate if it starts at beat 1 or 4; if it starts at beat 2, a quarter + an eighth is more natural:
X: 1 M: 6/8 K: C L: 1/8 CEGFDB,|CEGz3|CEGFDB|Cz2zEG|A6||
Generally, dotted rests are used in metres with three-beat units when the whole unit is silent. So you will see dotted quarter and dotted half rests in 12/8 metre.
You aren't likely to see things like a dotted half rest paired with a quarter note in a 4/4 bar.
I wouldn't call it a hard and fast rule, but this is how it's typically handled.