Yes you can. Here is a little link to help you out.
https://itsfoss.com/intsall-ubuntu-on-usb/
I didn't use this method, I have a persistent Linux install on a USB, but at least it'll point you in the right direction to figure out what you want to do.
You can use ubuntu, debian, or a few other live distros with "persistence" which is relatively easy to configure. This is neat because live versions usually come with cool features like auto configuration of devices and displays. You can even create a persistence boot of live isos on ventoy, which is honestly what I'd recommend.
I think you can simply install a Linux distro on a USB drive. You should use something fast like a USB-C hard drive and you'll have to think about where to put the boot loader. But if you're careful, what you have in mind should work.
I can't name specific solutions off the top of my head as I haven't done it in a while, but Yes. There are many live distros that offer an optional 'overlayFS' (I think that's the right term) which reserves some of your portable drive's storage for persistent changes to apply over the base live FS.
Combine that with git and/or NextCloud or an SSH file mount or three and you could probably come up with a complete mobile setup that also has quick access to whatever home server infrastructure you need. 128GB or 256GB USB sticks are pretty cheap now which is plenty for a spacious install. Or even a small USB SSD.
Don't use USB stick, it has really awful random read / write performance. I recommend fast SSD with cache. I tried USB stick solution several years ago and it was so laggy it was unusable
I inherited a crappy laptop (4 core atom processor, with 4 gigs of RAM, fear the power!) because Windows was running slow on it. I decided to try different Linux distros booting from a USB and had no issues. I literally ran the system for months off of a Sandisk USB drive, and it was faster than the spindle drive in the machine.
My recommendation is, don't cheap out on the USB drive. No-name drives are fine for word files, but the performance increase from a Sandisk, Samsung, Kingston or equivalent is worth it for any media transfers and will work fine for a bootable Linux.
Yes, definitely SSD is better but not as small as a little USB drive -- depends what you ultimately want :)
A good workaround for this is to add toram to your kernel command line. This loads the whole image into RAM before booting, which speeds things up dramatically at the expense of using more of your RAM while idle.
Yes. Installing linux in the first place involves putting it on a flashdrive and booting into it.
Full disclosure though: it wont exactly be the fastest system on an external drive.
Absolutely doable. I have had an Ubuntu on a 16Gb stick for years that used any time I had to log into my gmail from a borrowed/friend/relative computer.
It is doable to install an OS onto a flash drive or external drive, but from my experience it was really slow. Just need to make sure to then boot the machine to the USB device. Some machines you might find it difficult to change that in the BIOS.
Ha - I actually have Windows (WinToGo) dual booting with Ubuntu Budgie on the same usb stick - plus a non-encrypted ntfs partition. The OS partitions are encrypted though given that is portable. Working out the LUKS details and boot partitions was difficult however as there is a serious lack of good information on how to do it successfully.