Is the loss from a power transformer the same under load and no load?

Answer 1

No. The losses are more under load than no-load.
The losses in a transformer are grouped under two categories - fixed losses and load losses.
The fixed losses, as the name implies, remains more or less the same irrespective of the load. These losses are mainly in the magnetic core and due to the cyclic change in the magnetization of the core material.
The load losses are due to the heating effect of current flowing through the conductors of the windings. These losses are proportional to the square of the current and hence to the power flowing through the transformer.
The efficiency is given by the ratio of the total losses and the power input. Since a part of the losses is proportional to the square to the power, the efficiency should decrease with the increase of power, i.e. load.

Answer 2

No.
Transformers are probably the most efficient devices, with efficiency going close to 90%, but they do have losses.
The losses are essentially through Joule effect (the heating of the wires inside).
So, the more load you have, the more current you have, and the more the losses.
The efficiency, however, stays pretty much the same, even with the losses!

Answer 3

I would think that the greater the load on the transformer, the closer it is to saturation and more heat generated would lead to increased losses.

Answer 4

Maybe