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Object will not "experience" or feel time dilation itself.. but we observers will see the time for object slows and eventually stops, when the object reaches event horizon.
As for speed of fall into BH.. imagine the opposite: If object starts to move from event horizon away from BH with c (ok - ALMOST c..) the gravity of BH will slow it down, until it stops infinitely far away (We also have to imagine that in this universe there are no other objects).
The escape velocity for BH at the event horizon is c
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