What are some of Quan's BnBs?
b+2,1 > dash > b+2,1 > dash > b+1,1 xx Trance > b+1,1 xx mid Ground Burst
b+3,1+2 > dash > b+2,1 > dash > b+1,1 xx Trance > b+1,1 xx mid Ground Burst
u+3 xx Trance > b+2,1 > dash > b+2,1 > dash > b+1,1 xx far Ground Burst
f+1,2 xx Trance > b+2,1 > dash > b+2,1 > dash > 2,1,2 xx far Ground Burst
1,1,2 xx Trance > b+2,1 > dash > b+2,1 > dash > b+1,1 xx far Ground Burst
Trance > b+2,1 > dash > b+2,1 > dash > 2,1,2 xx far Ground Burst
NJP > dash > b+2,1 > dash > b+1,1 xx Trance > b+1,1 xx mid Ground Burst
air-to-air punch > dash > f+1 xx Trance > b+2,1 > dash > 2,1,2 xx far Ground Burst
After any Trance, you can instead do 1,1 xx Sky Drop for a reset. Keep everything in mind when deciding whether or not to do this: both players' meters, health bars, positioning, damage being forfeit if you choose to Sky Drop, etc.
After any Trance, you can either choose to gain meter by whiffing a Trance and a Ground Burst or, more easily but less rewarding, a Ground Burst and a Sky Drop, or perform a jump-in punch for some extra damage. You can also go for a mix of the two by whiffing a Trance and doing a jump-over punch, but off anything other than a max-distance Trance, it's a bit strict and requires that you switch sides with the opponent, since you have to whiff a neutral jump punch as they pass under you, then do a jump-in punch and continue the combo. Whiffing a Ground Burst is an easier method of doing this, but you build less meter and still have to switch sides except off max-distance Trances.
After any Sky Drop ender, you can perform the Rune Trap:
2,1,2 xx EX close Ground Burst > dash > 2,1,2 xx EX close Ground Burst (repeat)
The unblockable EX Ground Burst provides enough time to dash up and repeat the Rune Trap, or continue pressure with most of Quan Chi's strings (u+3 is too slow to be guaranteed). The only way out of this trap is to breaker unless Quan Chi is too slow. If, while doing the Rune Trap, the opponent doesn't block, you can choose to hit-confirm 2,1 xx EX Skull Ball, which staggers the opponent and gives Quan Chi a safe jump, but does about 3% less damage.
I usually just do what I'd normally do mid-screen, but replace Ground Burst enders with close Ground Burst. However, you can, off certain starters (b+2,1; NJP; f+2,1 xx Trance; Trance; etc.) that have not yet juggled the opponent, perform an additional b+2,1. (e.g. b+2,1 > b+2,1 > b+2,1 > b+1,1 xx Trance... or f+2,1 xx Trance > b+2,1 > b+2,1 > b+2,1 > b+1,1 xx close Ground Burst) Be sure you can do this consistently, however, since dropping it often ends with Quan Chi in the corner and swinging in the wrong direction as the opponent wakes up behind him. Additionally, you can replace b+1,1 xx Ground Burst in any juggles with 2,1,2 xx Ground Burst for a slight increase in damage.
Which of his strings are safe on block?
Not certain about the frame data, but judging from jumping with both characters after blocking:
f+1,2,1+2 is neutral on block and pushes most opponents out of d+1 range
1,1 is neutral on block, but the first hit can be crouched under
1,1,2 is neutral on block and pushes the opponent back, but the first hit can be crouched under
2,1,2 is -1 on block and pushes opponents back
2,1 is neutral on block
b+3,1+2 is neutral on block and pushes opponents back, but Quan Chi can be hit out of it between hits by pokes, uppercuts, and other fast specials/normals
b+2,1 is neutral on block
Which of his specials are safe on block?
Skull Balls are safe on block, but can be crouched under (about -2/-3 at point-blank range)
Ground Burst is safe on block against a large number of characters at farther ranges, but at close-range, I believe it's punishable by pokes and other fast normals/specials.
What are his worst matchups?
IMO: Cyber Sub-Zero, Johnny Cage, Kabal
What are his best launchers?
b+2,1; b+3,1+2; and 1,1,2
u+3; the 1+2 in b+3,1+2; and the u+3 in 2,1,u+3 (almost never use this last one)
EDIT: Forgot to mention that b+2,1 is neutral on block. Also, you can end combos with 1,1,2 xx Trance(whiff) if the opponent is grounded or b+1,1 xx Trance(whiff) if the opponent is being juggled for a bit less damage and a bit more meter.
EDIT 2: Also just realized you can replace b+1,1 xx mid Ground Burst with 2,1,2 xx far Ground Burst in mid-screen juggle combos. It does slightly more damage and might be easier to land, but also moves the opponent further. However, b+1,1 xx Trance(whiff) is still better if you want meter since the opponent recovers much more quickly from 2,1,2 than b+1,1.
EDIT 3: When doing b+3,1+2, if you're playing someone decent, you won't be able to catch them with the overhead the vast majority of the time, but in the case that b+3 is blocked, a connected 1+2 mid-screen can only be followed up by dash > b+2,1 > dash > b+1,1 on a few characters(Kung Lao, Quan Chi, and a couple others), so be ready to follow-up a connected 1+2 instead with either:
dash > b+1,1 xx Trance > 1,1 xx Sky Drop or
dash > 2 xx Trance > b+2,1 > dash > 2,1,2 xx far Ground Burst.