Your bicycle frame has a traditional BSA bottom bracket shell based on the link you provided about your current BB, which means you'd want to switch it for a BB-BSA24R-style BB. These are widely available for little money; Shimano for example has a budget option available as BB-RS501.
Random link: https://www.aliexpress.com/item/2251832565113180.html (select Color: RS501 BSA)

The store that sells the PR3 crank has an even cheaper version (the above link has extra shipping): https://www.aliexpress.com/item/3256805811238798.html (Color: B-BSA-24(68 73MM))

You'll also need tools to remove the old BB and install the new one. I don't know what tool is needed for the old one but it's likely an ISIS 20 tooth spline socket, readily available on AliExpress. For the new one you'll need 16-notch 44mm bottom bracket tool, also available all over the place (typically in either a cup or a wrench configuration).

And for the crankset you need a crank arm cap tool, but that's likely included with the crankset itself.

Regarding the question of just keeping the square spindle, that certainly seems like an option as well, especially if the current BB has plenty of life left on it. One question is how well the chainline would line up, I don't have any experience there.

One general concern is whether your frame can accommodate wider tires - what tire size do you think you'll be able to fit?

Most action cams have a time laps feature with pre-configured intervals. As a random example, this is from the GoPro Hero13 Black spec sheet:
Time Lapse Intervals (Video & Photo)
0.5, 1, 2, 3, 5, 10, 30, 60 sec.;
2, 5, 30, 60 min.

(I am not aware of any camera that would allow you to define your own interval (say, every 7 minutes) if that is what you are after.)

If you loosen both the expander plug and the stem (at the fork end), does the sound go away (or change)? If yes, it would speak to bits of the headset cap and spacer stack rubbing against each other or the frame.

Firstly, make sure that whatever bottom bracket you end up buying matches your exact T47 standard; SRAM for example lists these different options:
T47 (Road & Road Wide) 68 mm
T47 (Road & Road Wide) 77 mm
T47 (Road: 85.5 mm)
T47 (Road Wide: 85.5 mm)

Looking at Kactus vs. ZTTO specifically, the former uses ceramic bearings mostly while the latter uses steel bearings. Ceramic bearings are roughly 2 to 3 times more expensive than steel bearings, hence the price difference between the two. High-quality ceramic bearings perform marginally better than high-quality steel bearings; it's highly doubtful though whether your typical AliExpress bottom bracket contains high-quality bearings of any kind, be they steel or ceramic. Which doesn't mean that you shouldn't expect some useful life out of them.

Both ZTTO and Kactus are reasonably established players, making them reasonably safe bets. Whether they offer a better cost-to-performance ratio than SRAM (or any other manufacturer) is anyone's guess.

If it were me, I'd try to isolate the faulty component as a first step. I can't quite make out what you are saying in the video so ignore this if you've covered these steps already.

First, remove your rear wheel and the brake pads to avoid contamination with mineral oil; next, disconnect the brake hose from the caliper. Then try to move some mineral oil through the caliper with your bleed kit. If you can't, you know the brake is faulty.

If the path internal to the caliper is clear, either the hose is blocked (unlikely) or the hydraulic module within your shifter is borked.

A total noob question: What kind of software are you using to design stuff like this? Some sort of CAD-software? Is it free/open source? I’d be interested to design a storage box to mount inbetween the water bottles in the frame triangle on my TT-X38.

Yes, CAD / 3D-modeling software. Commercial products such as Fusion 360 are very slick and can do a lot of thinking for you; licenses for personal use may be available. On the opensource side, FreeCAD* has been around for a good long time but takes some getting used to, depending on how your brain works. Either way, expect a steep learning curve.

* Ondsel ES is built on top of FreeCAD and to an extent takes the sting out of FreeCAD's somewhat spartan approach.

I believe you are on the right track. Whether an axle conversion kit consists of just end caps or end caps with an axle largely depends on the design of the hub. In other words, some hubs are designed in such a way that all you need to do is switch the end caps; other hubs will require more surgery.

Some hubs don't offer any conversion kits at all so you need to carefully research what options are available for the hub model you're interested in. I'd go with hubs from a well-known brand like the ones you mentioned as they'll offer a high likelihood of a successful conversion. As far as the conversion kits themselves are concerned, I am guessing the ones made by DT Swiss are pretty pricy so finding a generic AliExpress substitute is the preferred option.