An alternative is always to have the Ottomans subsidize pirate havens in the Caribbean, which then over time become Ottoman colonies.
The reasoning for this could run thus: 1) Spain is our enemy -> 2) Spain gets a lot of its gold from the New World treasure fleet -> 3) Disrupting the New World treasure fleet will hurt Spain -> 4) Let's pay for pirates to disrupt the treasure fleet!
It doesn't even have to be Muslim corsairs - I'm sure there are more than enough Europeans who'd jump at the opportunity for state-sponsored piracy. So you could have a few Caribbean pirate havens, all nominally swearing fealty to the Ottomans, with Ottoman gold covertly funding the whole venture from Europe.
Then to have the Ottomans get Mexico and North America it's probably just a question of having a Drake-esque pirate do so much damage to the Spanish treasure fleets and, by extension, to the Spanish treasury, that the whole Habsburg Empire just goes bankrupt and collapses in on itself, leaving the colonies juicy targets for official Ottoman expansion, especially if Spain and Austria are no longer barriers to the Turks.
But no, this would not make the Ottomans stronger. Getting the Americas will just put in the Ottomans in the same situation as the 16thC Spanish - way too many fronts for the resources they have. Gold they invest in the New World is gold not being used to secure actual Imperial borders, and this will have repercussions.
As for names... probably Karayipler, Antiller, Meksika, Amerika. It's not like the Turks had an aversion to European names or anything.