Am I understanding the equivalence principle correctly?

It is the principle of equivalence that states that a uniform gravitational field and a uniformly accelerated frame of reference are equivalent. It is incorrect to say that gravitational time dilation and relative motion time dilation are the same thing since they are both results of different phenomena. One is a result of relative motion and the other is a result of gravity. Relative velocity time dilation/length contraction are an effect caused by relative motion. Gravitational time dilation and length contraction result from the fact that space-time is distorted in the presence of a gravitating object.

Because gravity and acceleration are the same thing, does that mean that the time dilation when near a gravitating mass is the same time dilation as when you are at a very high speed?

In some ways, yes. However, remember that what causes time dilation near a $g$-mass is the difference in $g$-potentials rather than $g$-acceleration. $g$-potential per unit mass has a dimension of velocity squared. In special relativity, what rules time dilation is the instantaneous velocity squared as appears in the Lorentz factor. I refer you to Einstein's words as he was extending SR to GR by finding some similarities between the centrifugal acceleration detected by an observer located at the center of a rotating disc and a corresponding $g$-field. You can also refer to this SE-question.

And does this prove that gravity and acceleration are the same thing?

Yes, at least locally.

Because gravity and acceleration are the same thing, does that mean that the time dilation when near a gravitating mass is the same time dilation as when you are at a very high speed? Are these two things connected/the same?

The principle of equivalence relates an accelerated frame and a gravitational field. Of course, as you said, it is only valid locally in the gravitational field.

All laws of physics work locally in the surface of earth, as work at a spaceship running with an linear acceleration $g$.

For example clocks ticking slowly, compared with an inertial temporarilly comoving frame in the case of the accelerated one. Or comparing with an object that is just starting to fall in a gravitational frame.

When there is a high relative speed between 2 frames, time dilation is symmetric, each one measures clocks ticking slowly at the other frame.

This is not the case of accelerated frames. Both frames agree about who has slowly clocks.

Because gravity and acceleration are the same thing, does that mean that the time dilation when near a gravitating mass is the same time dilation as when you are at a very high speed? Are these two things connected/the same?

No. In fact concerning GPS satellites SR effects and gravitational effects can compete each other. During the fist launches of GPS satellites, Russians have chosen orbits in which the two effects have compensated.

From this document from the Perimeter Institute

Einstein's theories of special relativity and general relativity have opposing effects on time in the GPS. Einstein’s theory of special relativity states that the clocks inside GPS satellites run slower than a stationary clock on Earth by 8.3 x 10-11 s per second. This is due to the speed of the satellites. Einstein’s theory of general relativity says the satellite clocks also run faster than those on Earth by 5.2 x 10-10 s per second because Earth’s gravity is weaker at the satellites’ altitude.