First a caveat: An object with mass can’t move at the speed of light, but it could move at speeds arbitrarily close to that.

The most successful model of gravity isGeneral Relativity, which treats gravity as a curvature of 4-dimensional space time. Gravity’s influence travels at the speed of light. There’s a classic thought experiment that sort of answers your question: what would happen if the sun was teleported away? The answer is the earth would continue to orbit around the spot the sun was for 8 minutes, and we would continue to see sunlight for that same amount of time since that’s how long it takes light to travel that distance. Then after 8 minutes the sun would disappear and the first “lack of gravity” would reach us, and things would be bad for earth :(

The fact that gravity travels at the speed of light actually leads to an interesting phenomenon: Gravitational waves If a massive object rapidly accelerates (or decelerates), for example a star sized mass moving quickly and then coming to an abrupt stop, it will emit a ripple in space time called a gravitational wave that will travel outward at the speed of light.

It was big news about a decade ago when gravitational waves were first detected by LIGO, a series of large interferometers that look for expansion/contraction in spacetime. Their first detection was the collision of 2 black holes; as the black holes spiral around each other and eventually merge, they emit oscillating waves with increasing frequency. They made a cool video showing how the frequency increases by converting it to sound.

Since then LIGO and VIRGO (similar European collaboration) have detected multiple gravitational waves from the collision of black holes and neutron stars. So not only are gravitational waves a neat validation of general relativity, they’re actually being used to do astronomy.

Using your choice of words it would be "stable/static". Effects of gravity moves at the speed of light. Perhaps a better example would be Earth orbiting the Sun.

The Earth is 8 light minutes away from the Sun. Meaning, the sunlight we see on Earth at this exact second left the Sun about 8 minutes ago. If we wave a magic wand and make the Sun blink out of existence in a fraction of a second, the Earth would continue to orbit the, now non-existent, Sun for the next 8 minutes. After 8 minutes the Earth would stop its circular orbit and head straight out of the solar system at what ever direction it was traveling at the end of the 8 minutes.

First, no object could be accelerated to that speed. Relativistic effects make that impossible. However, gravity waves move at the speed of light so there is some delay in gravitational effects. I'm not a physicist, but I'm pretty sure that if your sun-sized object shot through the solar system at 99.9999% the speed of light, and passed between the Earth and the Sun, it would take about 4 minutes for the object's gravity to be felt by either the Earth or the Sun.

Gravity is "emitted" by an object with mass. So to use what might be a better example, if a massive object popped into existence at a particular place, it would start "emitting" gravity waves from that time. Another object one light-minute away would start feeling its gravity about a minute after it appeared.

According to GTR, energy density bends spacetime, yes you can have an imaginative sun-sized object moving at c, bending spacetime, bc it is made of photons.