I had the same problem for the last 3 days and finally the issue was resolved, so here are some pointers:
1- Using Paypal standard payment worked but not the express checkout option, I was getting same message (6) Couldn't resolve host 'api-3t.paypal.com'. I was having the same problem with the UPS shipping and USPS shipping modules. In a nutshell every time I was trying to reach an outside server, it was failing
2- I have my own vps server and have many websites, and the problem was the same on all websites so it was not site related but server related.
3- Looking around forums, I finally found a temporary solution which consisted of changing the scripts in question. For example for UPS in the file includes/modules/shipping/ups.php on line 388 I replaced
if ($http->Connect('www.ups.com',80)) by
if ($http->Connect('153.2.228.50',80)) which is the IP address of the UPS server.
Note that this is a quick fix that will not resolve the larger problem at hand.
4- The real problem was that there has been an update of cpanel that created the problem and my hosting provider kept on ignoring that fact ( they usually wont admit to a problem being their responsibility). After 3 days of bugging them, they finally found the problem and solution on their end and ALL the problems disappeared as they appeared, instantly!!!
So basically if you run into this kind of problems it is most likely a server related problem. I am posting this hoping it will help some people from going thru the same loss of time I did and because often on these threads people don't post what the solution of the problem was (which is frustrating).