Why did the Gate closed when the SG-10 Team opened it from their side but did not close when homebase dialed into the black hole planet? Supposedly its drawing power from both sides, it should have the same effect right?
I believe the in-episode explanation was that when SG-10 dialed, the black-hole was still forming, and hadn't quite exerted enough force/provided enough power to the gate to have it stay open 'perpetually'.
Perhaps it (the Stargate) passed some sort of threshold where it started absorbing direct power (enough to sustain itself) from the black hole between these two events .
I'd say the simple solution to this is "time". While I think I remember them saying the Stargate might be somehow powered by the black hole, I'd just blame time dilation for this. I don't think it can be a power problem alone due to the 38 minutes rule.
I assume that the wormhole only collapses if both gates are inactive and without power/turned off, because some energy seems to be transferred in addition to travelers (there have been episodes where they arrived at a gate that hasn't had any working power supply).
So once they turn off the Earth gate, the "Black Hole" gate should shut off too. However, this doesn't happen immediately anymore due to dilated time, which causes the wormhole to stay active for a "bit" longer. In a similar way the 38 minutes rule doesn't apply, because for the remote gate this timespan hasn't passed so far.
In the first case (SG10 trying to get home) the effect isn't strong enough to be noticeable (other than in the form of the screwed up transmissions), but it changes later on.