Adapt the story to your players
You know your players better than anyone else. You should know whether they will react badly to their character being injured or dead.
Most players are fine with their character's losses as long as those losses can be recovered: the stolen money can be stolen back, the broken limb can be healed, the dead character can be resurrected... this is a common hook for additional adventures. But maybe your players are not like that.
Some players are fine with unrecoverable losses because they provide with good role-playing opportunities, but this is not as common, especially if the result is that their character does not seem as "useful" as the other characters in the party.
The most important aspect of a role-playing game is having fun. If in doubt, ask your players how they would react if "hypothetically, there was an accident and some of you lost a limb for some time...".
Make the story logical and believable
If you were the guard of a city and saw a carriage speeding towards your gates with a terrible monster on top, what would you do? I would say you would close the gates and fire your cannons against the monster. If that kills or maims some people in the carriage... well, your duty is to protect the city, not the carriage.
Most players will not feel as bad if they think that the bad outcomes of the story were the logical and kind-of-inevitable outcomes of the story.
I would say that the answer to your first question is "probably not", although this depends on your player as I said.
Recovery is a powerful motivator
For most players, the prospects of recovering something they have lost (stolen treasure, a limb, a friend's life) is motivation enough to do anything in the story: travel to a remote place to find the right cleric to resurrect your friend, enrol on a boat to know who stole your most precious jewel, etc.
I would say that the answer to your second question is "probably yes". Then again, this depends on your players.
Give them a little advantage but keep everything in the open
Many players do not like to feel railroaded into anything, much less into something bad for their characters. A common antidote for this is keeping all rolls in the open.
You can help your characters feel a bit better if you give them small advantages. For example the Hommonculus may have a penalty to dodge the cannonball because of its size, which means that the cannonball can be both a blessing and a curse. Another possibility is allowing a perception roll to the characters to see that the guard is reacting to them, aiming with a cannon, etc. If they do something sensible to avoid the cannonball, they may get a bonus to their dodge rolls, etc.
In short, giving them a little hope of avoiding the bad outcomes will make them feel better. Keeping all the rolls in the open will (probably) make them feel that you are fair.