There's also AmarecTV. Its kinda hard to set up (if you dont know what you are doing) but after you get it running it can record/stream easily and capturing wont take alot of cpu power.
http://www.amarectv.com/english/amarectv_e.htm
There's also VirtualDub but it's more about recording. You can preview the footage on this program too
http://www.virtualdub.org/
You can watch your own gameplay on amarectv/Vdub but there will be small delay (and the amount of delay depends totally on your capture card). But I'd recommend buying active splitter for quality recording/streaming (especially for very intense gaming and quality is always important)
Also the video feed from ps2 -> svideo is 480i so if you play through capture card video feed you will need to deinterlace it on the fly (built in amarec/vdub)
1. Here's one example of good active splitter which has svideo
http://www.amazon.com/Cable-Electronics-AV-400SV-Distribution/dp/B0009QZPO6/ref=sr_1_2?ie=UTF8&qid=1380780120&sr=8-2&keywords=celabs+s-video#_=_
so you will run ps2 -> svideo/rca audio -> splitter -> tv AND Capture card
+ some rca/svideo extension cables for your capture card and TV
2. Of course you can do it with Y-cable splitters but you will lose some guality <.<
I couldn't find any female svideo to 2x male cable splitters but i think the link will give you idea about it (good luck finding one)
1x http://www.amazon.com/30S2-12101-S-Video-Splitter-Female-Cable/dp/B001TPS63I
2x http://www.amazon.com/Monster-ILJRY-1-Single-Female-Y-Adapter/dp/B00006346W/ref=sr_1_7?ie=UTF8&qid=1387822440&sr=8-7&keywords=rca+splitter
so you will run ps2 -> svideo/rca audio -> 3x Y-splitter mess -> tv AND capture card
+ some rca/svideo extension cables for your capture card and TV
3. Also if buy one of these
http://www.amazon.com/Cable-S-Video-PS1-PS2-PS3/dp/B002NZTHEU/ref=sr_1_2?ie=UTF8&qid=1387824113&sr=8-2&keywords=s-video+ps2
You can run S-video to your capture card and composite (yup the horrible quality one) to your TV
here you will lose some quality as well but this solution is simply cheapest solution with capture card/TV solution
+ some rca extension cable(s) for your capture card and TV
(I personally cant play through capture card video feed)
you can always use line-in pc plug and just listen game audio through it
About recording you can use default codecs for it but there's much better option available (for example "lossless" codecs like Lagarith or Huffyuv). They take very little disk space and quality is great compared to other codecs
http://lags.leetcode.net/codec.html (I recommend this one. I'm using it)
http://neuron2.net/www.math.berkeley.edu/benrg/huffyuv.html (tried it -> ran into problems -> moved to lagarith)
I'll try to edit this post if I remember something vital :G
and I will answer questions ofc