If the strength of gravitational influence exerted by a body is derived from its mass and energy then is it true that a moving object which has some kinetic energy should also produce stronger gravitational pull than a static/slower one?
If that's correct then would it be also possible for an object moving fast enough to produce gravity so strong that it would collapse into a (fast moving) black hole?
No, you can't form a black hole by moving faster. The source term in general relativity includes mass-energy (rest energy) kinetic energy other energies and also momentum and stress. So when you just go faster your momentum increases too which affects the curvature too.
What does the curvature look like? If you moved fast near a massive body the massive body would curve spacetime a particular way and that curvature would look a certain way to you. Maybe unsurprisingly the fast moving body curves spacetime to look like that.
