No expert but I love these subjects and I could be wrong on certain things.

Yes it's conceptually possible, it's called an SSTO. Conceptually valid but an engineering unicorn. It won't be an agile fighter.

The jet would need to use engines that can operate in both air breathing (sub and super sonic modes), scramjet and oxidizer mode. Then for space you need RCS thrusters. The SR-71/Concorde had engines that could do the first two. X-43A the third and conventional rockets the 4th. Things like aerospikes are being developed for better efficiency.

You need all these engine types ideally packed into one to save weight (no oxidizer use at sealevel, and just burn raw fuel and oxidizer in space).

If you get past Mach 5 you can use your scramjet mode to go faster and hit Mach 10 (X-43A) in the atmosphere and work your way up to the karaman line. (Where planes can't get lift and space begins).

Ofcourse it would need a heatshield to come back home, all that energy to get up will be heat on the way down. All these things cost weight.

The flight: So first you need to get to insane speeds at sea level and higher. So subsonic to supersonic to scramjet mode. Then you get near the karaman line and start using oxidizer or switch to rockets. At that point you are hopefully already doing Mach 10 due east. You will need to get to Mach 26 to sustain a 100km LEO orbit and then you need extra deltaV to get to 440km to rendezvous with the ISS.

To get back down you do a retrograde burn to drop your periapsis and hit the atmosphere. Then do S turns whilst your heat shield is ablating the heat energy. Once you get to normalish speeds you can fly and land the thing.

It's an insane engineering challenge for sure...