Could you keep a star running for an indefinite future if you could keep pumping the accumulating helium out and fresh hydrogen in?
My take is that separating the helium out would be the hard part. It’s formed in a vast region; it’s not like there’s a simple cloud of nearly pure helium [here], while over [there] is where the pure hydrogen sits. Yes, over a long time the helium slowly collects in greater concentration towards the center. Maybe just stick your pipe in there at dead center and pull out a helium-enriched mix of whatever all’s down there.
No need for a Sun sized star. Just build a red dwarf. Say a third the mass of the sun. Those things will run forever. Long enough that the current age of the universe will be a mere infantile memory before they reach old age. Trillions, not billions of years.
Not in a red dwarf; they have convection currents such that they don’t run out of hydrogen until it’s virtually all fused. That’s part of why they live so long (the other part is that they burn dimmer).
So we build our own star. How do we access its energy? We could use something like photovoltaic panels to capture light from that star and convert it to electricity. Or we could let it heat up the atmosphere and cause wind, then build generators to convert that wind to electricity.