Eggs – How much oil is necessary to fry/cook eggs

First, about the temperature. Your safest option is to use a gun ;) The correct temp for shallow frying is between 150 and 190 degrees celsius. So if you have an infrared "gun" (a thermometer which neasures the temp of the surface at which it is pointed), use it to determine the stove setting at which the temperature of the dry (not ptfe coated) pan stabilizes in this interval after longer heating. If you don't have it, use the typical tricks for determining the pan temp, they are described in other questions here on SA.

Important: different stoves heat to different temperatures. Don't let the numbers at the knob to fool it into thinking that a given temp is "medium high" because it has the number 7. My own stove stabilizes at ~180 degrees C at setting 2 (out of 10), with setting 3 pushing 230. On the other hand, the minimum temperature for cooking fish is 50 deg. As stoves rarely go below 80 degrees, there is practically no chance to set your stove to a temperature so low that the fish stays undercooked (but it might need unreasonably long times on very low settings).

As for the spraying: the fat at frying temperature will always spray. You can still reduce your problems. First, the bubble explosions at the lower temperature will be less violent. Second, surface moisture promotes spraying (which is produced by water being instantly turned into steam in the oil). So pat the fish dry with paper before frying it. If this isn't enough, you can also drag it through flour. This will make the crust crisper. As the spraying won't be eliminated completely, don't throw in the pieces. Take a flat spatula with a long handle, place the piece on the working end, step back from the stove and slowly lay the fish into the oil, gradually pulling the spatula from beneath it. The turning should produce less spraying, so using a fork or spoon to press the fish against the spatula isn't a problem (or kitchen pincers, if you have those).

All of the above applies to everything you are shallow frying, not only fish. Only the flour is unsuitable for some vegetables.

My favourite oil for basting is none: I just put a glass lid on the pan, add a teaspon of water and let the top of the eggs cook in the steam. The few times I have basted, the olive oil I've used for cooking the eggs has not seemed to run off very well.