Are the game systems compatible? I was considering playing but I would prefer to play in the old republic era rather than the rebellion era. I haven't bought the other two games because I wanted to wait to play a game centered around the force.

Short answer, yes.

Slightly less short answer, the three different cores focus on different aspects but work well together within a group. Force and Destiny will be essential when playing in the old republic era to play jedi characters in my opinion.

Yes, they're all designed to work together, main difference is each one comes with different enemies in the cores, fluff that better describes different aspects (you won't really read much about smuggling in Age of Rebellion, or the resources of the Rebellion in Force and Destiny), and careers that better fit each one. But they're all designed to work together, and include tips for how to combine group mechanics (Duty, Obligation, Morality).

As for playing in an Old Republic era. By default, the era for all of them is in-between Episode IV and V, with a lot of general Rebellion era stuff. But most of the stuff can just be re-named or re-fluffed to fit the era, and you'd maybe only need to make very minor adjustments to adversary statblocks to better fit weapons they would carry in the era. The only real effort you'd need to put in would be statting out ships from the era.

The only real effort you'd need to put in would be statting out ships from the era.

Which, again, can most of the time just be re-named (or "re-skinned"): TIE fighters become "Sith Interceptors," and so on.

I'm starting an Old Republic era game soon myself, in the same continuity as the SWTOR MMO (with the characters as operatives for Sith Intelligence, working for Lana Beniko). Everyone is fine not using Force characters, so I'm just reskinning whatever I need from Edge of the Empire and Age of Rebellion: careers, ships, weapons, vehicles, etc. The Spy/Infiltrator career in particular is perfect for a SWTOR-type Agent Operative.

Not really on topic, but related: I plan to make Sith and Jedi present in the setting, but not abundant (definitely not as abundant as they are in the MMO). I want my PC's to be wary when they encounter someone wielding a lightsaber, regardless of faction.

I'm doing this sort of thing as well, and the three are absolutely interchangeable. I'm letting characters play species, enter careers and train in specializations in all three books. The only advice I'd give would be to have all players operating off of one set of character sheets. In my case they'll all be on Age of Rebellion, because of the mechanics unique to that game. The three are compatible, but you're better off only blending them in terms of character creation, not after that point.

Also, you can just re-skin any vehicles. I'm using the following conversions:

Ground Vehicles

Galactic Republic

Manka-class walker – AT-AT

BA-2 combat walker – AT-ST (new design, but walkers are referenced in the Battle of Corellia)

T-14 Skyspeeder – T-47 Airspeeder (remove tow cable; Battle of Corellia again; airspeeders appear in concept art)

Sith Empire

Jurgoran-class crawler tank – AT-AT (crawler tank seen in game)

Mailoc-class siege walker – AT-ST (siege walker seen in game)

Yozusk-class Airspeeder – T-47 Airspeeder (remove tow cable; speeders for balance)

Space Vehicles

Galactic Republic

Valour-class cruiser – MC-80 star cruiser

Thranta-class corvette – EF-76 Nebulon-B frigate

Wanderer-class transport – CR-90 corvette

Rendaran-class assault shuttle – Sentinel-class landing craft

Defender-class light corvette – YV-929 armed transport

XS-stock light freighter - YT-1300 freighter

Liberator-class starfighter – T-65B ‘X-Wing’

Equalizer-class bomber – BTL-A4 ‘Y-Wing’

Flashfire-class interceptor – RZ-1 ‘A-Wing’

Demolisher-class strike fighter – A/SF-01 ‘B-Wing’

Sith Empire

Harrower-class dreadnought – Imperial-I Star Destroyer

Terminus-class destroyer – Vindicator-class heavy cruiser

Gage-class transport – Vigil-class corvette

Imperial assault shuttle – ­Sentinel-class landing craft

D-5 Mantis Patrol Craft - Firespray Patrol Craft

Mark-IV Dominance-class starfighter – TIE Fighter

Mark-VI Supremacy­-class starfighter – TIE Interceptor

B-28 Extinction-class bomber – TIE Bomber

Mark-VII Eliminator-class attack craft – TIE Defender

The new fighters were created because I don't like the designs used in Starfighter. I hope this helps!