What happened between Halo 4 and Halo 5

Solution 1:

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 :(