Going to another star would be feasible if you did all of the above to a captured asteroid. A captured asteroid, 10km diameter (which is pretty BIG, and massive) could provide enough space and shielding for an interstellar journey (you could hollow out the inner 9.5km worth of asteroid and still have a radiation/micro-meteroid shield that was 'only' 250m (800ft) thick of solid silica-aluminate. You could is the ablated material of the asteroid as propellant for an electro-propulsive rocket system. Rotate the asteroid for gravity on the inner surface of the shell, etc...

A sustainable generation ship would probably require DD fusion. You just couldn't carry enough fuel to do it with fission or DT fusion...

The problem is coming up with an asteroid that big in a captured orbit (i.e. captured by Earth or possibly the Moon), and then supplying the necessary ∆V to break the capture and start the thing on its way....