Well, when I refer to power, I'm talking processing power not battery life. The Tegra K1 processor is one of the more powerful chips in mobile devices these days, which the Nexus 9 has. That said, everything you listed as doing is more than being capable of being handled by either device.
The K1 shines in the gaming arena, so if you're not really into 3D games so much, either tab would work happily for you.
The 16:9/4:3 aspect ratio is based around reading 8.5x11 paper, which is what most PDF's are modeled around for printing... 4:3 fits better with 8.5x11 paper when sideways, where 16x9 doesn't quite.
I don't remember if the stock ROM on the Samsung tab supports application installation onto SD card or not, but I know the custom ROM I have does.
The Tab's resolution is definitely better however it uses a PenTile display which some people don't like. Funny enough, the 10.5 however doesn't user PenTile and instead uses proper RGB.
http://www.gsmarena.com/samsung_galaxy_tab_s_105-review-1097p2.php
Tab uses AMOLED where the Nexus 9 uses IPS... Personal preference but most reviews I've read suggests that IPS is more accurate where AMOLED tends to wash colors out a bit.
About the ROMS and rooting - You have basically 4 "areas" of your device.
Recovery Bootloader (This thing allows you to flash ROMS. They come with a stock Bootloader by default, but you can root your device and flash a custom recovery like TWRP)
Custom ROM (This is the OS more or less of the device - what menus look like, how things work. When I talk about Lollipop, this is what I'm talking about)
Kernel (This is the interface between hardware and the ROM)
User data (Your stuff, like apps, games, etc.)
When you root, you're unlocking the bootloader allowing you to load custom "unsigned" ROMS. Often when you load a ROM, it'll have it's own Kernel which gets flashed as well.
So knowing all of this, to answer your question: Yes - You can download a ROM that has been taken stock from another device, and has been designed to "fit" on this hardware. In most cases, custom ROMs tend to have added features or additional abilities that stock ROMs won't have as well.
Lollipop is the name for Android v5, where older versions like v4.4 was KitKat, 4.1-4.3 was Jellybean, 4.0 was IceCream Sandwich, and so on. It's just a name for the general OS that was developed by Google, and then other manufacturers customize it to their device.
The mechanical keyboard depending on how you connect it shouldn't take up much power. I use a bluetooth keyboard on mine and the drain is pretty minimal - I guess it also depends on how much you use it. Either device (Nexus or Tab) will support it.
At the end of the day, if you're not into gaming as much, and the screen / Rooting is more important, go with the Tab S. If you're into gaming and need lots of horsepower so to speak, the Nexus 9 is a monster. Either will suit your needs, but for me the only reason I went with the Tab S 10.5 instead of the Nexus 9 was because the 10.5 was larger and I do my gaming on my PC.
Hope this helps :)