Q. Is it right to assume that those 8 (or more) non-scholarship players are mostly recruited?
A. Yes. Nearly 100%. The number of non-recruited walk-ons who make D1 rosters is vanishingly small.
Q. Are all places required to hold open tryouts?
A. No. Some states and some schools require them, but there is no NCAA rule requiring it. Even where open tryouts are held, it is extremely rare for a player to show up at an open tryout and make the fall roster. When it does happen, it's most often a potential bullpen catcher. Bear in mind the open tryouts usually occur around the first official practice day in mid- to late September, by which time the recruited players (both scholarship and non-scholarship) have already been working together for several weeks, going through conditioning drills, skill work, four-on-ones, and player-led practices. It's nearly impossible for a non-recruited player to show up at the open tryout and gain entry to this already well defined unit.
Q. Do recruited walk-ons sometimes lose roster spots in early tryouts?
A. Recruited walk-ons don't participate in the open tryouts. They're already on the fall roster (even though won't get posted until after the open tryouts). A lot of times, the open tryout occurs on the first official day of fall practice--but after the team has finished its practice or scrimmage. The recruited walk-ons participate in the practice, not the tryout.
Assumption: I assume up front a recruited walk-on won't get much playing time.
Assessment of assumption. Often true, but there are lots and lots of notable exceptions. Coaches make scholarship investments based largely on their talent evaluations. They are open to having non-scholarship players prove them wrong, but it's not easy because the coaches are usually right.
Q. What are the odds of a recruited non-scholarship player not ending up on the roster at all?
A. It depends on the school. Many D1's start fall practice with between 38 and 45 players. There could be as many as 18 or 20 recruited walk-ons competing for as few as 8 roster spots. At D1's that aren't fully funded, there may be only 32 or so players on the fall roster and everyone who stays eligible and out of trouble can make the spring roster. At a very few very competitive D1's, they don't make the final cuts until a few days before the spring season starts, after it's too late for the last players cut to transfer to a JUCO or a D2. Most players at most schools know their status at the end of fall practice.
Q. How often does a recruited walk-on remain in that status for multiple years? Do most quit or get pushed aside after, say, a year?
A. Regardless of whether they are scholarship or non-scholarship players, most D1 college players have a limited time to convince the coaches they are ready to help the team win. Attrition is high for both groups of players at most D1 schools. Look at the recruiting history for schools your son is interested in. A school that brings in 18-20 players a year can sustain that level of recruiting only if half or more of each class doesn't come back for their sophomore year. School that bring in 10-12 each year can sustain that pace with normal, unforced attrition.
Best wishes,