A quick calculation tells us that a quasi-star with a radius of 10 billion kilometers (a figure suggested by the Wikipedia page) at 4.25 lyr (Proxima's distance from the Sun) would have an angular diameter of 1.7 arcmin, which is about three times Jupiter's.
As a comparison, here from left to right there are the apparent sizes of the Sun, the quasi-star and Jupiter.
Since we are able to take exceptionally detailed images of Jupiter, even from the ground, then we would definitely be able to study the surface of the quasi star in detail.
As for the last question, yes, it would be like a second Sun. A star with that radius, and a temperature of 10000 K (from Wikipedia) would have an apparent magnitude of -23, while the Sun has a magnitude of -26.7.
Most of the quasi-star's light would be in the UV, and so it would be blocked by the atmosphere, but it would emit in the visible enough to lighten up the night.