Physically I think such an object is possible. I would not know how a planet could gain such a high rotation velocity.
In the solar system massive planets tend to have a shorter rotation period than less massive planets (10 hours in the case of Jupiter).
Earth would have to spin with 11 km/s (escape velocity) for experiencing weightlessness at the equator resulting in a rotation period of 1 hour. The surface of Earth would be in geostationary orbit. A planet spinning with escape velocity though would be unstable and break apart.
Jupiter would have to spin 60 km/s for centrifugal force to equate g-force. To get 1 g at the equator rotation velocity would be something like 50 km/s or over 80% of its escape velocity resulting in a rotational period of 2 1/2 hours. That is only 4 times faster than Jupiter spins already.
Of course in reality Jupiter would lose all the hydrogen and helium by atmospheric escape, resulting in a smaller but denser planet, a sort of super-earth, so all parameters would change.
The form of a fast spinning planet would be ellipsoid rather than spherical.
We have flattened objects in the solar system, for example Haumea in the Kuiper belt. It rotates with an equatorial velocity of 400 m/s which is almost 60% of its escape velocity of 700 m/s.
Dwarf planets with rotation velocitiy approaching escape velocity obviously can form, although they are pretty rare. If and how a fast rotating super-earth - that is what you are asking for - can form I do not know.
"Would spaceflight be as much possible from the surface of such planet as from Earth?"
Yes, from the equator.
"Is it possible that at the equator ... the effective gravity is moderate (say, 1 g) but on poles the gravity is so huge that nobody can walk there?"
A super-earth with a radius of 2 earth radii would have a surface gravity of over 3 g (says google) and would have to spin with close to 90% of its escape velocity to get down to 1 g at the equator, pretty close but - theoretically - still below break-up velocity. At the poles you would still experience 3 g, which means it would feel like you carry a bag on your shoulder twice the weight of your body. I do not think I could walk there.