To start off, a good guide to many quastions on continuo realization 
is:
North, Nigel. Continuo Playing on the Lute, Archlute and Theorbo. 
Bloomington: Indiana University Press, 1987.
See especially Chapter 5, "Cadences, Sequences and Unfigured Basses".

> In a minor key, I'm not entirely sure what to do with the V chord. 
> When there's *no other* indication to tell me what kind of third to
> play (e.g. a melody note, or some other written part, or a sharp sign
> in the figures), does the V chord take a major third or a minor third?
>  In a major key, obviously the V is major, but what about a minor
> key...?

In a minor tonality it depends on context. Simply put, if the V chod 
acts in direct relationship with the tonic chord, especially if it 
preceeds the tonic chord, the third of the V chord is major.

> And, for that matter, is the third in any 4-3 (11-10) suspension
> *always* a major third?

Often; not necessarily; it depends on context.

> I'm looking at Caccini's "Amarilli mia bella."  One flat, but I'm
> assuming that the key is G minor.

Yes; one flat was a standard way of indicating G minor (actually 
Dorian) in the seventeenth century. In hexachord thinking (which 
Caccini and others of his generation would have been taught - after 
all, when he was a young boy it was within the Renaissance), the 
sixth degree of Dorian (E or E flat in G Dorian) could be flat or 
natural depending on context.

  Bar 2 begins with an 11x10
> suspension over D in the bass:  does the "x" mean a major 3rd?

The X is a sharp, so yes. (The seventeenth century did not use double 
sharps.) What to us looks like an X was one of the standard ways of 
writing the sharp sign. 
For examples, see our online facsimiles of seventeenth-century books 
at:
http://ace.acadiau.ca/score/facsm/ctcc/#
http://ace.acadiau.ca/score/facsim5/caccini/contents.htm
[for the second, start with "Che fai mifero coreecco ch'in Croce"]

  If
> there were no "x" there, would you still play a major third?

Possibly.

> Bar 3 begins with the note D in the bass with A in the melody.  What
> about the middle note?

Open to interpretation: it could be F# or F natural.

  Assuming D is the dominant (assuming the key
> is G minor), does that mean that it always takes a major third?

More often than not, but specific context would be a determining 
factor. 

For this particular song, an interesting almost contemporary source 
that has a written out lute part in tablature with the voice is 
Robert Dowland's Musicall Banquet, 1610. It provides at least one 
view of the appropriate resolution of the continuo.

For Le nuove musiche, it is informative to look at H. Wiley 
Hitchcock's modern edition, published as part of A-R Editions series: 
Recent Researches in the Music of the Baroque Era, vol. 9. 
See: http://www.areditions.com/rr/rrb/b009.html
It is out of stock, but any decent music library will have it.

GJC


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