Since when do phasers draw blood?
I've seen every episode of the original series and all of the original-cast movies, and I don't remember phasers ever drawing blood before this movie. They either "stun" or they kill by dematerializing the target, depending on their user-controlled setting. If I remember right, in some of the early TOS episodes, the "kill" setting of the phaser was shown to kill without dematerializing the person, but there was still no blood nor specific wound to be seen.
Even in this movie, when ersatz Saavik demonstrated the phaser on the cooking pot it dematerialized it rather than putting a hole in it like it did to the Klingons.
Even if you suppose that the phasers have a setting that makes them work like a plasma cutter, there should still be little to no blood, since the intense heat would inherently cauterize the wound. Plus, why would assassins choose such a setting? Dematerializion is obviously the most effective setting, by far.
Also, what happened to those wide-spread torpedoes that were so successful against cloaked ships in the original series? Those also existed in the Star Trek arcade game (Sega, 1982).
Plus, what's so difficult about targeting a cloaked ship after it's fired on you? It should be a simple matter for the ship's computer to calculate where the shot came from; it's just a straight-line trajectory. They are even watching the shots approach on the view screen, no less. The 23rd-century computer doesn't know what to do with that information? I can see it being a problem if the cloaked ship is doing strafing runs at warp speed, but that wasn't happening in this movie. It was just sitting there taking pot shots at the Enterprise.
Of course, if Star Trek were written in a way that makes sense, no one would ever get hit with those slow "dramatic" torpedoes, regardless of whether or not the attacking ship was cloaked. Sensors would detect their approach as soon as they were fired and the ship, which can reach faster-than-light speeds in the blink of an eye, would simply move out of the way. And it wouldn't wait for an incredibly slow-by-comparison captain to give the order; the computer would make the decision in a matter of nanoseconds.