Any American citizens recruited by the Btitish army at that stage would have been volunteers, not 'pressed'.
The fact that they were volunteers is probably why America didn't go to war, but the fact that recruiting them is still a breach of neutrality legislation is probably why Britain accepted the dismissal.
Only part of the Royal Navy. Those ships which were not involved in the Black Sea operations would have been more than sufficient to eat the entire United States Navy for breakfast and then wonder what to do for lunch.
The crisis came to a head after the Crimea was over so the availability of ships historically is not a perfect indicator, but the Times on 11 June 1856 indicated that, once proposed reinforcements had been dispatched, the North America station would consist of the following ships:
Nile (91)
Powerful* (84)
Boscawen* (70)
Imaum* (72)
Pembroke (60)
Cornwallis (60)
Euryalus (51)
Amphion (34)
Vestal* (26)
Eurydice* (26)
Termagant (24)
Cossack (21)
Pylades (21)
Arachne* (18)
Malacca (17)
Falcon (17)
Archer* (15)
Mariner* (12)
Daring* (12)
Buzzard+ (6)
Argus+ (6)
Basilisk+ (6)
Scorpion* (6)
Hermes+ (6)
(Note: * = sail, + = paddle)
It's not the worst fleet in the world, but it's qualitatively and quantitatively inferior to the force Admiral Milne had during the Trent affair (which included nine screw battleships) and lacks big frigates capable of taking on the USS Merrimack- though, fortunately, she's the only one of her class which would have been ready. It's also far short of the force the Admiralty Hydrographer thought Milne would need to just blockade the northern coastline (6 battleships, 11 frigates, 23 sloops, 20 gunboats). This is where the issue of the Crimea comes in: it becomes more difficult for the Royal Navy to raise such a force when ships and/or men that in 1862 would be available for North America are being used against Russia.
Notice that the US wouldn't be doing this out of the goodness of their heart, but for an issue of neutral rights and potential gain of territory. In essence, the United States would be throwing the dice in the same way they did in 1812 and might have done in 1862. It's a gamble that they can take and hold Canada quickly enough that the British can't reinforce it; that the harm British naval power can do them is minimal or at least tolerable; and that the stalemate will force Britain to come to terms in such a way as to leave the US with Canada at the peace table. I'm making no prediction of whether those assumptions are correct or not, just pointing out that the American government might have thought it worth the stake.