I'd say "stake" is a general generic term for ownership by an individual or group of individuals in a corporate or other business entity; or a game or other enterprise.

"Stock" is a very specific type of financial instrument, controlled in the US by the Securities and Exchange Commission.

If a company has 100,000 shares of stock outstanding, and person A and person B each own 100 shares, you can say they have equal stakes in the company.  The actual value may rise and fall.  But they each continue to own one one-thousandth of the outstanding stock.

If, as in your example, the company issues and sells another 100,000 shares, the proportion of their stakes in the company will be "diluted," or in this case cut in half.  It would be like watering down their drinks.

1.  No

2.  In my experience, "stake" is most often uncountable,  but since each shareholder may be said to have a stake, five shareholders may have five stakes.  I think in your example they wish to compare the stake of the shareholders as a group, with the total value, or with the government's stake.  (both singular)

3.  "Shareholders" are always "stakeholders" but stakeholders in a poker game would not be called shareholders in the legal corporate sense.

I can make a private deal with five guys to finance my treasure-hunting expedition.  They each give me $500, and I promise each of them ten percent of what I find.  They are equal stakeholders, but I get half.  You could call them "shareholders" if you wished, but not "stockholders;" and though you might be able to sue in civil court if I don't keep my promise, I don't think I'd be guilty of a federal crime.