Misconceptions

Part 1:

Something raised to power 0 is 1 and here it is tending to $0^{0}$ . That is $0.000000000000.....1^{0}$ or negative of this, so answer is 1.

Part 2:

Limit means that x is tending to 0, not 0. So answer is $0^{(not 0)}=0$ .

Part 3:

We can see that moving close to 0, we get 0.001^0.001 which approaches 1. This limit can be evaluated by writing as $e^{xlnx}$ and applying LH rule for exponent.

Part 4:

$0^{\infty}$ is always 0, whether 0 is exact or tending.

$0.0000001^{\infty}$ is 0 as for x belonging to (0,1) the answer is always 0.

$-0.00000001^{\infty}$ is also 0 as for x belonging to (-1,0) also the answer is always 0, so the limit reduces to 0 as both right and left limit are 0.