I agree that the gastraphetes suits dwarves, and that true crossbows should be a later (and mannish) invention.
As for varied types of "Tatical Styles", I confess I'm no expert. I dabble in military and technological history, but my focus has always been political. So I'm only going to be able to give vague impressions... but that's never stopped me before!
This might not be to everyone's taste, but I think Morgoth's Angband military style has very little nuance or cleverness. He does have an active R&D Department, to borrow Corey's phrase, but it's more about making bigger and better monsters. He is clever and diabolical in his dealings outside of battles, with Hurin and Gondolin, and in gathering allies among the Easterlings, but his battle strategy seems to be "throw thousands of minions and dozens of very heavy things at my enemies"; he's more Machiavelli than Sun Tzu, at least to my mind. Morgoth should be, however, patient and very good at making sure his pieces are all in place before battle -- again, Machiavelli -- and he should never attempt to attack before he is confident he has the upper hand in firepower. But in battle itself, formations could be loose, or even fall apart completely once a charge begins, and orcs could fight more like individual warriors than a collection of soldiers (think Gothic "barbarians" versus Roman legion). This will mean they fall in greater numbers, but could also make any maneuvers they do pull off harder to detect. In orcish culture, a mindset could even develop that hiding behind shields, or attempting to rely on formations/your fellow soldier is weak or cowardly, encouraging orc to attempt even more rash solo attacks, to prove themselves to their peers and higher-ups, wrenching their way up the social ladder of Angband -- after all, who wants to be on the bottom of that?
Elves, in contrast, should quickly develop tactics for being outnumbered, sometimes overwhelmingly so. A focus on formations, support troops, and mobility makes sense for them. A dependence on one another could set up how devastating the betrayals at the Nirnaeth will be. On the other hand, this dependence and discipline would allow them to execute maneuvers like Hannibal's men did at Cannae. In other words, it would work against the orcs beautifully, if not for the Curse laid upon the Noldor...
Men, at least the Edain, could mimic elven styles of warfare, until the fallen Easterlings begin to fight more like hordes of orcs.