There is a Kansai Thru pass offerred by Nankai. It's 5000 yen for 3 days travel
http://www.surutto.com/conts/ticket/3dayeng/index.html#about

Does anyone know if this includes travel from Osaka to Kansai Airport? The website only says that it can be used when arrive in Kansai Airport to Osaka. What 'bout the opposite direction when I'm leaving Japan.

Also, how convinient is it to use this pass to travel around Osaka & Kobe region? compare to JR rail pass of course.

Cheers

Yes, airport access by Nankai Railway is covered (in both directions), but not airport access by JR. If you travel by Nankai limited express (as opposed to express), you will have to pay a supplement even when using the Kansai Thru Pass.

Also, how convinient is it to use this pass to travel around Osaka & Kobe region? compare to JR rail pass of course.

I'm planning to go to Koyasan as well. I did a bit research and it seems that the only train to Koyasan is Nankai and the return ticket is 2780 yen (correct me if I'm wrong). So I guess by adding the airport ticket which is 1390 yen, it is ok to purchase the 5000 yen ticket.

Btw, what is Nankai limited express? Is it just another type of express train on some of the routes? so, bascially I can wait for an express train instead on any of the Nankai line. Is it correct?

Note also that you can use the pass not only on Nankai but on most trains and buses in the greater Osaka region, except JR trains.

Btw, what is Nankai limited express? Is it just another type of express train on some of the routes?

It is a faster category of train, for which you have to pay a small supplement.

so, bascially I can wait for an express train instead on any of the Nankai line. Is it correct?

I looked at the link you provided and it says:

''The ticket can be used on any two resp. three days within the period of validity, except the Zenkoku Version of the 3 Day Ticket, which is only valid on three consecutive days. ''

Does this mean the 3-day Kansai Thru can be used for any 3 days? Doesn't have to be consecutive days?

However, there are many, many Hikari connections between the two cities (~2 trains per hour), which are not that much slower. You can use the Hyperdia website to check your train connections:

http://www.hyperdia.com/cgi-english/hyperWeb.cgi