Linux

Rufus is open source, and you can view its source code here. If you have any reason not to trust it, you can audit the source code yourself. All things said, it's bot a very complex program. Auditing it yourself wouldn't take a ridiculous amount of time.

Linux distros are not going to distribute .exe files. First off, those are binary files and you cannot verify the integrity of the binaries the same way you can verify the integrity of an ISO. An ISO has all of the files packaged in a readable format, and you can view and verify them manually. Second, compiling a .exe would require the use of Windows, which is non-free software, going against the ideology of open source practices. It is unnecessary and quite frankly ridiculous. A Linux distro is complicated and difficult to maintain as is; each distro being required to maintain their own Windows version of the installer adds a significant amount of work to developers that (for the most part) already aren't getting paid. The developers working on Linux distros do not have Windows installed, and thus will have greater limitations to testing the installers in Windows. Binary files will work differently on different versions of Windows, which will only introduce more complexity and difficulty to maintaining such an option. Additionally, it would add compilation time to create the .exe files, which will cost money and time over the long run. There are already open source and well trusted softwares to create liveUSBs from ISOs in Windows, it's just impractical and unnecessary for distros to maintain and distribute those tools themselves. Besides, as far as your "trust" situation goes, that just shifts trust to another entity; it's zero sum anyway. Just use a tool that already exists like Rufus, Balena Etcher, or Ventoy (all of which are open source).

As far as answering the rest of your question, all you need to do is download the ISO from your distro's website, download and install Rufus, Balena Etcher, or Ventoy (your choice, doesn't matter which), plug in a USB device with at least 8GB that you can use for the liveUSB, run Rufus/Etcher/Ventoy, select the USB device, select the ISO file, hit confirm, and you have a liveUSB. Now all you need to do is reboot your computer and (for most systems), it will automatically boot into the live OS where you can install through a GUI installer. In some systems, you'll need to boot into the BIOS/UEFI settings and enable or move USB boot up in the boot order, or manually boot from USB in the boot selection menu (the button to enter the boot selection screen varies by manufacturer). Then it will boot into the live OS and you can click the install button when it pops up. It's actually almost the exact same process you'd use to make installation media for Windows and install it to another device, except Microsoft also has their own installation media creator that asks for the same stuff as Rufus if you don't want to use the ISO from Microsoft for whatever reason. You could also just use Rufus with the Windows ISO though, it's essentially the exact same thing.