How stable is a donut-shaped star or planet configuration and under what conditions such object may form? Is there any evidence suggesting that such objects might exist?
In principle, yes. In practice, no.
The question has been studied for a long time. The classic treatment is Dyson's papers on "anchor rings" in 1893 (paper I, paper II) but it goes back to the study of "figures of equilibrium" starting with Newton's considerations of the oblateness of the Earth and then continuing with Maclaurin and Jacobi's analysis of rotating ellipsoids of rotating homogeneous self-gravitating fluids (see Chandrasekhar's book Ellipsoidal Figures of Equilibrium 1969 for everything on the topic; a shorter overview).
The basic story for ellipsoids is that as you add angular momentum they become more oblate (the Maclaurin ellipsoids), undergoing an instability beyond a critical point to become general three-axis ellipsoids (the Jacobi ellipsoids), that may also contain fluid motions (the Dedekind ellipsoids). Eventually there is too much angular momentum and it breaks up, but Cartan showed that the mode that grows fastest is the low-frequency harmonic making it just split into two. So this way you will not get a torus-shaped planet or star.
Dyson considered the instabilities of a ring already existing. He found that for nearly circular rings:
The annular form of equilibrium of rotating gravitating fluid is stable for disturbances symmetrical about the axis, and for disturbances which alter the shape o,f the central curve, but is unstable for long beaded disturbances.This result was, perhaps, to be expected, as by means of beaded waves, the mass would naturally be broken up into spheroidal masses.
Similar results seem to hold for more ellipsoidal rings.
The condition for instability is $R>3r$, that is, thin hoops are unstable while really thick doughnuts may be stable.
On the other hand, an accreting disk may somehow shed angular momentum. Could that produce a fluid or solid torus (albeit with some body in the middle)? This is the problem of why Saturn retains its rings. Tisserand showed that $N$ satellites could orbit in the same orbit if $m/M < 2.3/N^3$ -- the more satellites, the more they would tend to clump unless they were very tiny. (Sheeres & Vinh 1993) tighten the bound a bit. So unless there are other forces than gravity keeping things in place a dust or gas ring will tend to clump.
Since then others have studied the question, such as (Wong 1974). They found that in principle it looks like there is a pathway of unstable intermediate shapes from the Maclaurin ellipsoids to the torii (see Ansorg, Kleinwächter & Meinel 2003), and if there is enough dissipation of the right kind it might counteract the beaded instability splitting it. But it all looks unlikely to happen unless everything is set up perfectly.
Wong's paper speculates in various astrophysical cases where a torus may be possible. The closest to reality is likely ring galaxies, where the stellar ring remains for a relatively long time. There is also a fair bit of interest in toroidal clouds around black holes (with or without accretion disks), since they make sense as models for active galactic nuclei. But here there are gas pressure and luminosity from disks complicating things: it is far from a ring-shaped star or planet.
(Which is a shame, since I had so much fun modelling them!)
No, such a configuration would not be stable. Matter tends to clump together and as the mass increases the rotation slows.
If the rotation were too fast for the material at any given size then the stresses would be too great for it to remain as a cohesive body. In these cases you would get something more like a rubble pile. If the rotation were somehow sufficiently high enough to theoretically form a donut shape, I think the gravitational force of the matter would not be enough for it to coalesce into any sort of cohesive body.
Consider, for conservation of angular momentum the angular velocity decreases as the distance from the centre of rotation increases. That is, stuff closer to the centre rotates faster than stuff farther away. But for a solid body the angular rotation is the same at all distances from the centre. This means that the angular velocity (in degrees per unit of time) is the same at all distances, and in this case the stuff closer to the centre rotates more slowly than the stuff farther away.
So to form a solid body the stuff in the centre needs to slow down as it accretes matter - it rotates more slowly as it gets bigger. This would prevent forming a donut because the centripetal forces don't outweigh the gravitational forces. Fluid bodies like stars or planets with liquid/molten cores are more complex but the same idea applies.
COnsider this counterexample: the asteroid belt. Taken as a whole it's toroidal, but only because all the components are revolving about the sun. They wouldn't even form a torus in the absence of their orbital parameters. Larger clumps (i.e. protoplanets) can only form due to gravitational attraction, which produces straight-line attractive forces (as opposed to, e.g., the path of a free electron in a magnetic field). All material is drawn towards a common center.
Simple answer: no.
Shapes like a doughnut (this shape is called a torus) can exist in space, but not in dense material. Dense material becomes spherical because gravity drags it together and a sphere is the smallest it can get (size depends on what is opposing gravity - turbulence or radiation etc.)
The definition of a planet actually involves spherical shape because this is related to size and mass. Asteroids can be made of similar material to planets, and be orbiting the sun, but one of the reasons we don't call them planets because they are not spherical.