1.Sirius is a possibility for a myriad of reasons but you are right it is a bit far to imagine we could orbit the center of mass (roughly 6.6ly) in ~24,000 years. Nonetheless, some suggest it is possible, including astrophysicist Reg Cahill of Australia who calculates the solar system is moving ~430km/s, almost the exact speed required. But he measures relative to the CMB and it is uncertain if this is the right methodology. Some also argue that parallax measurements might need adjustment by a factor of 2 to 4x but I have not studied this enough to comment. With current precession theory effectively constraining solar system motion to zero it may take awhile before these issues are seriously examined. Yes, Alpha Centauri appears to be closer but it is much farther off plane, whereas Sirius is closer to our plane and closer to the Pluto and Sedna inclines. It is interesting to note that Barnard's Star lies in the opposite direction, roughly in line, and it has the highest proper motion of any nearby star. Your questions about perturbations are difficult to answer until we know what is going on. Heck, no one is even certain what keeps Sedna in its current position - and that is much closer than anything we are now discussing.
2. I am skeptical of the brown dwarf scenario for exactly that reason (although it could be against the galactic center background where there is a lot of noise), but there are some very good scientists, including Mike Brown at Caltech, that are seriously looking for such an object, partially because Sedna demands the presence of another massive object.
3. It is hard to prove a negative (such as x object does NOT exist) but lunisolar theory has much bigger problems. Certainly there are some local effects like nutation but the bulk of what we call the lunisolar effect (sun moving across sky at about 50.3"p/y) is probably just a reference frame issue, and this is why there appears to be so little precession relative to local objects and so much relative to distant objects (outside the moving solar system).
Chinese Philosopher – Lao Tzu