I've finally got some soldiers who lived long enough to be high rank, and I've got some extra supplies laying around. I'm eyeing some of the Guerrilla Tactics bonuses like Specialist: Cool Under Pressure, which grants +10 aim and ability to crit and reaction and overwatch shots, but I haven't been able to determine if these bonuses which require a specific class to unlock only apply to that specific class, or if they apply to all soldiers.
No, only soldiers with that class receive the bonus associated with the perk from the GTS. Recruits receive no kinds of buffs until they rank up to Squaddie.
The bonuses granted by the Guerilla Tactics School are retroactive (bugged right now, as they only trigger on promotion, but the values set for each squad member is set when the AWC is built.), much like the Advanced Warfare Center, so yes, your soldiers that you had recruited before the GTS was built will receive those bonuses when you purchase them.
So if you had a Specialist before your GTS was built, then that Specialist will also get that bonus going forwards still. If your recruit didn't spec into Specialist, then they don't get the buff, as the tooltip references which class each perk works with.
As an example, I didn't get a GTS until late game, where I already had a lot of Colonels. I bought the Grenadier perk from the GTS that allows grenades to deal crit damage, and from that point on, even my Colonels and Captains could crit grenade enemies, despite them being along in my squad for far longer than I had the GTS built.