Have the Elites learned to speak English (without any automatic translators) by the time of Halo 5

Here is the review of the games up to Halo 3.

1. You wake up
2. Covenant attack the ship you are on
3. The general for the Pillar of Autumn tries to land on the Halo ring
4. You get out of the ship on an escape pod
5. You crash land on ring
6. You search for others who used pods
7. Go into the facility to find the Halo control room
8. Covenant accidentally release Flood
9. Work with guilty spark to kill Flood
10. You figure out Guilty Spark is trying to turn on ring
11. You self destruct the crashed Pillar of Autumn to destroy the ring

1. It shows a Covenant Elite commander being stripped of his rank and then tortured by Tatarus, a Brute warrior
2. The Chief, Johnson, and soldiers are awarded for their actions on the first ring
3. A Covenant fleet appears outside of Earth
4. They attack New Mombassa, Kenya.
5. Chief takes them out, and the Covenant retreat through slipspace
6. The Chief fallows in the In Amber Clad
7. They find another ring, and chief goes to destroy it
8. The disgraced Covenant commander is offered position of the Arbiter
9. He goes to kill a heretic, and finds 343 Guilty Spark (Oracle)
10. The Covenant go to the newly found ring due to a distress signal from the Prophet of Regret
11. Tatarus tells Arbiter that the Prophets plan on annihilating the Elites
12. The Arbiter is sent down a chasm
13. The Gravemind saves him, and the Arbiter meets the chief
14. The Gravemind tells Arbiter that the Great journey will wipe out the Covenant, Humans, and the Flood
15. The Chief teleported to High Charity where a civil war is happening
16. Cortona realizes that the Gravemind used them as a distraction to infest the In Amber Clad
17. The Gravemind crashes it into High Charity to spread the Flood
18. Chief goes onto Forerunner ship and leaves Cortona behind to destroy High Charity and the Halo ring
19. Arbiter goes to the ring and rallies his allies to fight the brutes
20. Arbiter and Johnson kill Tartarus
21. They turn off the ring, but instead it makes it so all rings can be activated from the "ark" at a safe distance
22. Chief goes back to Earth

Much like the other answer, your best bet is to consume the other media in the Halo Universe. Namely the Kilo Five Trilogy, as these take place after the main war of Halo 1-3 (some of it during the events of 4). But for the sake of saving time... I'll do my best to answer your specific points.

Osiris is one of the many Spartan IV fireteams, as Spartan IVs can be selected from "normal" soldiers (Like Buck) and upgraded, instead of taken at a young age and "created" like the Spartan IIs or the mass produced expendable IIIs. As for the members... Nightfall is a good explanation for Locke, and you should already know Buck if you played ODST... but what about Tanaka and Vale? For this, I recommend reading their HaloPedia entries Tanaka and Vale as they are certainly just minor characters, and are pretty much just there to fill the 4 player co-op, and match Blue Team.

That made a good segway into Blue Team, yeah? Well, if you've read Fall of Reach you should already know Blue Team and what happened to them... the Kilo Five Trilogy I mentioned earlier also follows along their endeavors while Chief is doing his thing, but again reading up on Blue team will certainly help if you don't want to read 3 reasonably sized books. Another book that covers Blue Team's survivors specifically is "Last Light"

As for Halsey's side of things... she's all over the place... and I think this is intended. There probably is media explaining exactly what was going on with her during the war and aftermath... but I haven't really come across much. Kilo Five again would be the best bet here.

I never made it to the end of Spartan Ops... so I don't know what exactly you are asking :(