I haven't read Persichetti’s book so I maybe don't understand the context of your questions well enough to give you a satisfying answer. That said, I have studied 20C music at length and frequently use all sorts of synthetic scales in my own compositions, so hopefully something I say will be useful.
Terminology might differ by region, but I think in general the term synthetic scale simply refers to any scale that is somehow an artificial construct of the mind, ie invented, rather than having originated "naturally" within a particular music culture (the way the diatonic church modes did in the West). This includes scales which are derived from traditional scales -- eg by altering notes or by combining diatonic tetrachords that don't normally go together -- as well as scales which are invented based on some sort of theoretical principle -- eg symmetrical scales like whole tone or octatonic, which are created by equally dividing the octave into repeated intervalic patterns.
This is a huge and heterogeneous category, so it is difficult to impart general guidelines for how to use them. Which scale are we talking about? What are you trying to achieve musically? Presumably you're not going for functional harmony, but maybe you still want a tonic? In composition, anything goes. The exciting thing about abandoning the rules of common practice music is that you get to make your own set of rules.
To become familiar with a new scale/mode, I suggest noodling in that scale on a polyphonic instrument such as piano or guitar. That way you can explore the harmonic possibilities. If you're interested in having a tonic, then I'd suggest that you drone the tonic while you noodle, because you'll want to hear how each note feels relative to the tonic. If you're really interested in harmony, I'd suggest being systematic.
How to handle harmony? I don't think I can give any generally applicable guidelines here. Literally any combination of notes is a valid chord. Some will be useful and some won't. In most cases, you're unlikely to find chords which will drive your music the same way dominant quality chords drive common practice music. But you might be able to achieve a pseudo-functional effect by resolving intervals (eg tritones) in traditional ways, only in the context of the new scale.
Do I still need to avoid the the Tritones for that scale in order to stay true to the mode?
I don't see why you would need to avoid tritones in any scale. Both tonal and atonal music and full of tritones. Melodic tritones are typically avoided in common practice music, but they're allowed -- even required -- by its harmony.
How do I find the characteristic notes in a scale
Every note in a scale is important. The idea of some notes being somehow "characteristic" of a scale only makes sense when you consider scales relative to each other. Take dorian mode for example. When compared against the natural minor scale its characteristic note is its raised 6th degree, whereas when compared against the major scale its characteristic notes are its lowered 3rd and 7th degrees. Simply put, distinguishing features depend on context. It might not be the scale degrees themselves which are characteristic, but maybe a particular intervallic pattern.
That's all pretty vague, I know, but it's a broad question. Hopefully you'll find some utility in my answer.
