For example 1 I would use a (big) Composite Time Signature of

{3 over 4+8} + {1 over 2+4} (that is right; multiple characters in one field)

This big time signature will (almost) give the desired beaming.

You can fix the beaming via

Utilities menu > Rebeam > Rebeam to Time Signature…

Add false barlines with the Expression Tool.

Be careful with note accidentals - since many barlines are “false”.

For the time signature display, choose the Time Signature option

Use a Different Time Signature for Display

- and set a “Display” time signature of 3/8.

Now the big time signature will display as 3/8 in both staves.

In the Viola staff, use the Staff Tool to make Time Signature an Independent Element.

Now you can make the Viola staff display a time signature of {9/8 + 3/4}

For example 2 it is unclear what you mean:

… create each time signature component as a separate measure (e. g. 3/2 would be measure 1, 7/8 would be measure 2, etc.) …

A) This sounds like the first four measures will be {3/2}, {7/8}, {3/2}, {7/8}.

… By default, all components of a compound meter is put into 1 measure. (3/2+7/8 is all one measure, creating 19 eighth beats, when I would like each consecutive measure to alternate between 3/2 and 7/8.) …

B) But this sounds like the first two measures will be {3/2 + 7/8}, {3/2 + 7/8}.

Which is it?

A) = four measures

or

B) = two measures

???

Thanks for your elaborate answer for my inquiry 1. (@Peter Thomsen.) I learned actually quite a bit from it. (Compound meters that create a complex denominator was not something that I had learned nor that had occurred to me before!)

~~~~~~~~

For inquiry 2, what I'm saying is that, by default, Finale produces B): {3/2 + 7/8}, {3/2 + 7/8}.

What I wanted to know was how to command Finale to produce A):  {3/2}, {7/8}, {3/2}, {7/8}.

Take a look at Joseph Martin's choral composition, "Come to the Music", page 1. 3293529-1.png (815×1632) (jwpepper.com) (Depending on the browser, you might have to zoom in.)

He too uses a compound meter: {6/8+2/4}. On the other hand, it's partitioned into two measures, where measure 5 contains the 6/8 while measure 6 contains the 2/4; continuing that alternating pattern for consecutive pairs of measures until the time signature is changed (which, on an irrelevant note, happens on page 3).

What Finale would do is put {6/8+2/4} all in measure 5 and every consecutive individual measure thereafter (until time change). For my own compound meter, {3/2+7/8}, I don't want this. I want what Joseph Martin has in his "Come to the Music" pg 1.

Now, you actually already indirectly provided one solution: to create false barlines with the Expressions Tool. I suppose that would work, but adding to the complication of having to assess accidentals, I would additionally have to deal with the hassle of measure numbers, and that is just way too much excessively unecessary work. (Eeeks!!)

Nevermind inquiry 2. I figured it out myself. 

Actually, you already indirectly provided another solution: "Use a different time signature for display."

For the first measure, I created {3/2}. For the second measure, I created {7/8}. For both measures, I commanded that Finale "[uses] a different time measure": {3/2+78}. Then, to save myself the hassle of repeating these steps for all subsequent measure pairs, I simply copied and pasted the original pair onto those subsequents until I had a meter change. Since all measures take on the 'false' identity of {3/2+78}, only the initial measure displays {3/2+78}.

Thank you again for your help.

Now, I'm assuming inquiry 3 is out of the question? Since you answered most of it, the one thing that hasn't yet been is how to incorporate the parenthesis (). Should I assume that there is no current Finale feature that can do this?

… Since you answered most of it, the one thing that hasn't yet been is how to incorporate the parenthesis () …

I would use the Expression Tool.

By The Way, a thought about your Example 1:

You wrote:

“… after 5 violin meters, both Viola and Violin's meters will synchronize again …”

Your words “will synchronize again” mean that the barline should be drawn through the staves after 5 violin meters (but not after the first 3 violin meters), to clearly show the overall pattern - right?

Yes ^^^^

That is precisely the idea.

I'm presuming you're hitting at some pleasantly clever idea yet to be shared? =)

… I'm presuming you're hitting at some pleasantly clever idea yet to be shared? …

Not really.

It is possible to have the barline drawn through the staves after 5 violin meters.

I explained how to do so in my first post:

Only one, single (big) “real” measure for the 5 violin meters.

Use the Expression Tool to insert false barlines.

It is also possible to have the barline drawn through the staves after 3 violin meters, and again after 5 violin meters:

Just two, (big) “real” measures.

First measure: {3 over 4+8}

Second measure: {1 over 2+4}