New answers tagged replacement
Those are sheet metal screws. They never fit very well and it is easy to strip the socket. Phillips head will usually do better than a slotted head as long as the screwdriver is a tight fit. You cannot normally get a hex-type head on a sheet metal screw, especially one that small.
That diameter would be very close to a M2.5 at 1.993mm minor screw. A hex head cap screw (AKA Allen head) could replace the Phillips. Its also a bit smaller than an ANSI #4-40 at .0805inch minor You could try the approximate size and get the matching tap for the closest interference fit.
I would use fig. 1. It would give the wall more strength on the side(s). When the new window is installed, it will be a more steady installation for the siding on the outside.
I suspect the round outer part above the water spout turns. However, it may be impossible to remove. Would probably be easier and less frustrating just to replace the tap. (Sorry.)
From my point of view (image is very small and poorly illuminated) you face a hexagonal ending of the shaft, when looking at the top of the tap. My first try was to get a wrench socket, which fits the hexagonal stub. It may be difficult to find a matching wrench socket small enough to fit into the housing of the tap, but you could take a cheap one and grind ...
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