How would bitcoin network react to big scale network failure? Let's assume the netwrok gets partitioned eg. computers in North America cannot contact computers from other continents. Would payments still be processed in separate partitions? What would happen after restoration of communication?

How long is the split?  There's really no problem if the split is less than the block generation maturation time (20+ hours)-- a bunch of miners will be disappointed on one side of the split or the other, but that's about the extent of the damage.

Longer than 24 hours... is kind of hard to imagine for a big country.

Would transactions continue to be processed:  yes, but...  the sudden drop in network hashing rate (and the drastic slow-down in block generation) might trigger future safety checks in Bitcoin, so you might have to do something special to tell it "yes, I really do want to generate transactions even though something weird is happening with the network."

If a little country or region got split from the main network, it will probably have a lot less hashing power and it will take much longer to generate the 100 blocks needed to start to cause problems.  That's a feature, not a bug.

After communication was restored the more-difficulty block chain would "win" and transactions from the losing block chain would get retransmitted and move over to the more-difficulty chain.

If somebody had a super-secret communication channel that worked during the split they could use it to double-spend.  But if the bad guys have a super-secret channel then probably some good guys would, too, and as jgarzik points out, it only takes one little link to relay blocks and prevent a split.