1) Does the tree in the photo look like a good anchor point?
It looks okay, but beware that it's a dead tree. Looks like the smaller branches are very dry and could potentially break, so don't use those as your anchor point. If you tie the rope around one of the thicker, major branches, it should be fine. Those still seem to be in good condition.
2) I want to use an anchor hitch to tie the rope to the branch. Is this knot reliable and safe?
Yes, the
Anchor Hitch is safe to use. To be extra safe, add a
stopper knot at the free end, close the Anchor Hitch. In other words, tie the stopper knot close to the last finishing
Half Hitch (which is the last step of tying the Anchor Hitch).
Alternatively, you could also add an
extra Half Hitch around the standing part, followed by a stopper knot.
3) If the noose will end up being too close to the ground what should I do with the excess rope?
I'm not completely sure what you're asking here exactly. What does the height of the noose have to do with the amount of excess rope?
Regarding excess rope at the free end... If there is a lot of excess rope, it will hang downward, and might reach the ground, even if the noose is high enough. It's not a problem, and you can just leave it hanging.
You could also tie the knots in a way that most of the excess rope is on the anchor side, in which case you can just wrap it around the branch or something to get it out of the way. However, if you do this, make sure you still leave
some excess rope at the noose end too. There should always be a tail left at the free end, which should be at least 10-15 cm long, or at least about 10 times the diameter of the rope. If there is no tail left or it's too short, there is a higher risk of the knots coming undone.
Regarding the height of the noose... If the noose is too low, too close to the ground, that will definitely be a problem, regardless of the amount of excess rope at the free end.
4) Should I pre-tighten the noose knot before stepping off?
Yes, definitely. The noose should be pre-tightened and positioned at the back of the neck.