Start a game with 4 players?
 

Due to illness, Team A only has four players available to start the game. The game may legally begin in this manner.

Is this legal? Are you ever able to START a game with only 4 players?

Not legal in Fed per 3.1.1 Note (emphasis added):

A team must begin the game with five players, but if it has no substitutes to replace a disqualified or injured players, it must continue continue with fewer than five. When there is only one player participating for a team, the team shall forfeit the game, unless the referee believes that team has an opportunity to win the game.

Do you have your books with you? Check out rule 3-1-1 Note, it details your very question.

No offense intended, but it seems like you're posting questions directly from the NFHS test.

NCAA is the same (3-1.1) "At the start of the game, each team shall consist of five players, one of whom shall be the captain."

Just to add a bit of a twist to this discussion...

Team A has only four healthy players. But for whatever reason they decide to play the game. So they submit their lineup before the 10 minute mark with the four healthy players and one unhealthy one. Their plan is to start the game, then withdraw the fifth player due to his existing illness or injury.

If you know this is their plan, do you allow them to proceed and "skirt the rules"? If, by the second half, the withdrawn player feels well enough to play, do you allow him to play?

Also, what if Team A arrives with only five players, all of whom is healthy. But after the 10 minute mark, one is injured. Do you allow them to start with fewer than five since they had five on the lineup at the 10 minute mark and it is permissible to change a starter due to "necessitated by illness, injury, illegal equipment or apparel, etc., or to attempt a technical-foul free throw"

"Only if they are playing the game on a cruise ship that is zipzaggingly traversing the International date line and at least one of the game officials is sporting fishnets and high heels....."

Leave it alone!!!!!!!

Yes, if that player is on the floor to start the game (i.e., for the opening jump ball...or FT's if a T is called).
If at the time of the jump (or Technical FT's to start the game), they can not put 5 players on the floor, they do not play. Forfeit.

Nope. You need five.

I had an interesting situation last night.

Team B calls timeout.

After the warning horn, B13 reports for B44. B13 enters the game. I inform the coach that she must wait and that B44 must remain in the game.

She said, ummm, well that won't happen because B44 just left she had a family emergency. Can we play with 4 until she can come in?

I said, no that's not necessary, B13 can sub in. And we continued from there...

+1 for exercising common sense.

Exacly. Injuries, illnesses, etc. trump other substitution limitations. Normally, it would involve a player on the floor but I can imagine situations to expand that to include others.

Exactly, even if you didn't know the rule, common sense should dictate here.