Soundcard suggestions?

I don't have the UR-22, though from all I've heard so far it is a very good sound device. I therefore cannot answer your question with 100% certainty. I can tell you however that I can route Windows' default audio to my Zoom R24 which is also an ASIO-capable USB2.0 device. So if we are talking about Windows and you are running at least Windows 7, you should be fine.

It should be pretty much the same deal on other systems. I can't speak from experience with regard to Macs, but the sound architecture of Mac OS is actually supposed to be a lot cleaner than that of Windows. On Linux there is also no difference, a sound device is a sound device (usually ALSA is used as hardware driver/backend nowadays). However, I'm pretty sure the Steinberg interface is not an ideal candidate for Linux; at least from what I have read, I would recommend the Focusrite Scarlett for that purpose. On Windows and Mac the UR-22 should perform admirably.

In general there is no difference between a 'soundcard' and 'audio interface'. The former is an older term from ye goode olde days when audio devices literally were extension 'cards' (i.e. ISA, PCI(e) or PCMCIA), while the latter is a bit more generic: since the development of USB (and FireWire), a lot of stuff that used to need special interfaces or extension cards in the computer now comes in form of external boxes. Nowadays PCIe audio cards are only used for special applications in science and engineering (where extremly low latency and jitter values might be really vital).

The Tascam would do the job just fine also (I have one, runnin on Win XP or Win7), and in fact, there are many, many other models that would satisfy your needs too; look at the offerings from Presonus, Focusrite, MOTU, RME to name a few...
Have a look here for example:
http://www.musicradar.com/news/tech/12-...s-570850/1
The good news is: I think that there are no more really "bad" interfaces on the market anymore! :-)

I use the ur22 and i like it. For that price, it's great.
I use to have the tascam 122 and it didn't work very well with win7 on my laptop.

Slightly, if you are lucky. In the Win7+ world, even 'native' ASIO sound cards quite often use kernel streaming in the background AFAIK, since the hardware driver model is actually quite restrictive. In other words, it used to be much simpler to get direct access to the sound hardware in the olden days. Nowadays you are expected to go through the 'proper' channels... which means that an ASIO driver most often is a thin wrapper anyway, and the manufacturer-provided one might be a bit faster... and it also might be a bit less stable (the Zoom R24's driver is an example of the second category).