Bumpy switch from Sony RX10 III to Fuji X-T4

You are really unlucky here ! Personally, I would return it as there are too many issues. However, I'll try to respond with what I would personally do for each line item.

hayenne wrote:

1. The 18-55 lens is weirdly blurry, a la tilt-shift lens, I am attaching a sample image. I am unable to take a single picture I would like with it. There is no optical damage as far as I can see, but when I tried attaching it to a couple of other bodies in a local shop, it gave the lens control error. The seller swears it did not fall and has been cooperative with providing receipts of purchase and generally not disappearing from the radar. The local shop suggested they can send it to Fuji and wait for them to fix it (out of my pocket since it's no longer on warranty)

The fact you have managed to test this lens on another body (good job) is a great way to isolate the issue. I would ask the seller for a refund based on this assessment, alternatively they could agree to pay for the repairs (if you trust the seller).

2. 35 f2 is somewhat noisy, I can hear the ambient whirring sounds all the time as long as it is attached. It seems to work otherwise well.

Is the sound evident when you are attempting to AF-C/AF-S ? What happens when you put it in Manual mode, do you hear the same noise ?

3. Is the biggest complaint actually - the battery discharges extremely fast. I took a camera for a walk with the 18-55 lens attached (before I realised it was faulty) and it lost 50% of charge over a matter of 1.5 hours and some 30 images. Naturally I switched it off when unused. At first I suspected OIS motor sapping all the energy, but yesterday I used 35f2 and battery lost 20% over some meagre 30 images again. This is nowhere near I expected it to perform.

I am using my Samsung USB-C fast charger (5V 3A) and charge in-camera since I suppose it makes no difference. The camera reports the battery is in excellent shape.

Have you tried performing a full camera factory reset and trying again ?

Here is a thread that is somewhat related (X-T3) - https://www.dpreview.com/forums/thread/4619686