First, I'm not clear about what you're seeing on the bottom of your fruits on the Beefsteak. If it's a dark leathery colored area that might have some crcaks in it, or even black in color, it's called BLossom End Rot ( BER), which is a physiological problem, not a disease, and if you go to the main forum, link at the top of this page and go to the FAQ area at the top there's a good blurb about BER.

(1) Will the green fruit ripen once it is off the plant and how long does it take?

Totally green fruits will not ropen up well. If it's the BER you're worried about that's a temporary condition and is usually seen on first fruits and goes away as the season progresses.

(2) What could be the reason Beef variety was cracking and may have been infested where as the "Ace" variety was fine?

Different varieties respond differently in terms of BER, some never, some a lot.

(3) Is there any way to accelerate ripening without fruit becoming too big or staying on the plant for long

Do you mean accelerating ripening while the fruits are still on the vine? Not that I know of and honestly, you really don't want to do that and again, if it's the BER you're concerned about you won't have to.

Hope that helps.

Carolyn