Singing or speaking when counting
Damping the string during rests
Maintaining a clear connection to the melody while playing
When to stick with it and when to move on
Pick direction in triple meters (9/8)

 You suggest in your book to practice counting out loud. Does it matter if it's singing or will speaking do?
Either one is fine. The main benefit is derived from invoking the part of your brain devoted to speech-- so any audible sound is as good as another. If you are shy about singing, or if someone at home is going to give you grief, it's perfectly fine to speak softly, or even whisper, but make a sound pass your lips. (The cross-picking/ chordal variations later in the book aren't especially singer-friendly, so speaking is definitely in order for those.)

 Should I damp the string at a rest?
Yes, damping the string through rests is a good habit to cultivate. It matters less in a solo context since mandolins don't have that much intrinsic sustain, but if you are playing chamber music, particularly baroque chamber music, you will want to damp the string so that the other voice(s) can be heard without clutter. (It may well be the case in chamber music that the chord changes during your rest, and if you let the note ring, it will clash.)

Also, it tends to be more difficult for many people to 'play' rests of the correct duration than to play a flurry of notes, so it's helpful to have a physical action to associate with the rest. Generally, you won't want to cut the note short, though, so count the note for its full value, then 'play' the rest by damping if the note hasn't already faded out naturally.

 When I'm playing the chordal "Ode to Joy" (exercise 5, p 24) my wife says it really sounds good but I can not hear the melody as hard as I try. What does that mean?
Are you able to hear the melody in the very basic version on p.21? The melody remains the same as the very basic version, actually. It's just filled out a little more so try and listen for the basic melody as you are playing. If you still can't hear it, record yourself and see if you are able to hear it on the recording. Most likely, your brain is a little overwhelmed with all the things you are asking it to do simultaneously since the skills are new. As you grow more accustomed to reading and playing at the same time (cognitively an entirely different skill than decrypting, memorizing, and playing what you remember), you will be able to listen to yourself more musically; it will help you to hear yourself more thoroughly as you play, and to add inflections as you play rather than playing a string of notes. That will come with (and, alas, only after) practice. Everyone experiences a mechanical phase while learning to read music. Keep practicing and the mechanical aspect will recede eventually.

Another thing that is likely to be instructive is to go through the chordal version and the basic version on p. 21. Compare them side by side, and circle all the notes on the chordal version corresponding to the notes on p. 21. That will highlight the melody on the page for you. Thereafter, as you play, if you add slight emphasis to the circled notes, it will then bring the melody out more than playing all the notes with equal emphasis, and you will be more likely to hear it distinctly. As you become more proficient, you'll be able to identify the melody notes without needing to circle them.

It's particularly important to keep your fingers down, especially on the embellished versions of all the tunes in the book: the melody note might need to ring through some ornamental notes, and if you let go of it, the melody has literally disappeared and all that is left is the ornamental accompaniment.

 Can you give me any advice on when to "stick" and when to "move"? Right now, I feel like when I finish the book, I'll start right back again at the beginning to keep myself sharp. Even a couple of days without reading and when I go back to it, it's not as easy. So knowing I'm going to come back to it, how "down" should I get each exercise before moving on?
Keeping in mind that the point here is to enrich your musical life and not to flog yourself with exercises, I would suggest sticking to it until you the piece provokes so much frustration that the exercise doesn't hold your interest. There is, of course, a certain amount of necessary diligence to push through mild frustration, so spend at least one week on each section in the book. Beyond that, if you feel saturated with frustration, it's better to put the piece aside for a while and come back to it later than to keep grinding away at it when it doesn't engage your mind. (There is plenty of music in the world. No need to labor over something your mind is tired of.)

The book is structured in sections, with exercises progressing in difficulty. If you find yourself getting tired of particular exercises in the middle of a section (after a one week minimum), skip to the beginning of the next section and come back to the skipped fragment later. Concentrate on one section at a time, however. Don't work on several sections of the book simultaneously.

As for things not being as sharp after even a few days of not reading, yes, that is an unfortunate reality of the adult mind. Use it or lose it definitely applies. Reading every day-- even without an instrument in hand-- will help you keep the skill. Ten minutes a day, every day, will stick far better than nothing all week followed by an hour on Saturday.

 I have a question concerning the right-hand technique for the Jigs on page 15. They are in 6/8 time and I have seen others play Down-up-Down, Down-Up-Down for each triplet. I believe this to be correct as it lends itself to the rhythm of the jig. You have written that I should use alternate picking consistently throughout the book. Should I make an exception here or do you suggest sticking to the alternate picking? I can learn either way but can you explain if one way is suitable over the other?
Regarding jigs (or any DANCE music in triple meters like 6/8, 9/8 etc.): yes, down-up-down, down-up-down is the convention; go ahead and use it as you practice if it doesn't throw you off. It pays to be disciplined about your pick direction habits. If it throws you off, just use alternate picking at first. Once the alternate-picking habit is well formed, it's easy to add another pattern, and you can add the down-up-down, down-up-down pattern quite easily, later.

The reason for the down-up-down, down-up-down convention is to put rhythmic emphasis on beats 1 and 4 (hence resetting your stroke to "down" on beat 4). Any time you are playing dance music, you will want to inspire a foot-tapping impulse in listeners who will hopefully then get up and dance, so use this convention in any time meter divisible by 3 (e.g. 6/8--jig, 9/8- slip jig).

However, if you are NOT playing dance music, heavy emphasis on beats 1 and 4 might not be appropriate and you may prefer a more homogenous, lyrical tone (either from alternate picking with a nice clear up-stroke, or from all-down-strokes). For example, "Jesu, Joy of Man's Desiring" (p. 28) is an excerpt from Bach's Cantata 147-- church music written for choir with instrumental accompaniment. You don't want to create foot-stomping energy there-- it's not a slip-jig at all but, rather, a lyrical accompaniment for the (rhythmically slower) main melody carried by the vocalists.