We arrive at ICN at 5:20 am and depart at 2:50 pm on Tuesday, Oct. 27. Is it reasonable to try to do the Yonggungsa Temple tour AND the Incheon Temple tour? The latter is from 9 am to 11 am. If it is feasible to do both, would it be better to sign up for the Yonggungsa Temple tour from 7 am to 8 am (Can we get there in time with a 5:20 am arrival?) or from 8 an to 9 am (Would we be able to get after that to the Incheon Temple tour leaving at 9 am. Does one tour end in exactly the same place as the next one begins? Do you need to check in in advance of the start time?)

Thanks in advance for your help.

Have you considered going into Seoul on your own? No need for a tour there unless you want one.

I've heard that it's quite expensive to go into Seoul on one's own, and hence was thinking of the free tours offered by the airport. Also, we will have just flown all night (from Hanoi) and probably won't be at our sharpest.

I thought the cost of a bus into Seoul quite reasonable -- about 12 or 13 USD each way, less if you take an "all-stop" train rather than an express. Here's some info:
http://asiaenglish.visitkorea.or.kr/...EN_2_2_2_3.jsp

Your call!

I can't speak to the tours as we went into Seoul independently during our own connections at Incheon, including last month and previously after arriving from Hanoi about the same time in the morning that you will be. Both times, we found getting through immigration/customs to be very quick--within 15-20 minutes from getting off the plane. So in response to your question about timing, if your incoming flight is on time, a 7 am tour should be feasible.

In addition to kja's link above, the airport website also has some useful info and maps:
http://www.cyberairport.kr/pa/en/a/index.jsp