The draft is 40 rounds long now, though you could in theory have more picks due to supplementary picks. I believe as it stands the Yankees are entitled to 42 picks in 2014, for instance (they gain three picks due to losing Kuroda, Cano, and Granderson, but lose one fr signing Ellsbury; that can still change, depending who signs where.) A team can stop picking before round 40. I believe there used to be more rounds; Mike Piazza was famously drafted 62nd solely as a symbolic gesture of kindness to his father, and then they kind of found out by accident that he could hit.
There is no minimum, but practically speaking a team could not possibly draft fewer than 20-25 players a year; they will lose that many players every year just to sheer attrition. Baseball’s minor league system is vast, because it takes years and year of pro ball for 99% of major league ballplayers to learn to play MLB ball. Every MLB team has 200+ players in the minor leagues and overseas affiliates under contract, and every year a lot of them give up or have to be released when it is discovered that they are simply not pro material. MLB teams chew though a huge number of players.
A point I should make is that to get a compensation pick for a lost free agent, a team must make a “qualifying offer,” which is defined as a one-year guaranteed contract for a number equivalent to, IIRC, the average of the 125 highest salaries in MLB from the year before. Again IIRC that was $14.1 million. So the Yankees had to give Robinson Cano an official offer of at least $14.1 million for one year (obviously a formality in his case since he would be insane to accept it) to get that draft pick. When you’re talking about big stars the qualifying offer is always a pure formality, but for players with question marks this becomes a tricky situation. Going back t the Blue Jays, they lost Josh Johnson to free agency and got no draft pick because they did not make him a qualifying offer. Johnson has been an excellent pitcher in the past but in 2013 was a horrible disaster and got hurt, so it was a risk; do you make the offer? He might accept. If he does, will you blow your $14 million or get a surprising year out of him? They chose not to risk it, electing to spend the money elsewhere, so he left and they get no pick.
As to why Phil Bickford turned down a $4 million bonus, it’s because he got bad advice. There is simply no other explanation; they thought the Blue Jays would blink, and that was a terrible miscalculation. Aside from the fact the Jays are owned by a giant, soulless media conglomerate that would happily drain the blood from its custodial staff if they thought they could make thirty bucks doing it, the new draft rules are specifically designed to punish the Phil Bickfords of the world by giving the Blue Jays every reason to tell him to go to hell. They had very little to lose by standing their ground, because now they just get another draft pick in pretty much the same position in the draft. For them it’s hardly even an inconvenience; for Bickford it’s potentially a life-altering mistake with little upside.
I suspect MLB was unofficially, er, supporting the Blue Jays in their stand, since it sends a rather stark message to future Phil Bickfords.