So I think I found an explanation on p. 90 sec. 9.3 of Turkish: A Comprehensive Grammar (www.eric.com.br/turkish-grammar/turkish-grammar.pdf), which suggests that ülke ülke is functioning as an adverb in this sentence, perhaps as in something like, "I'm traveling countrywise."
I think this must be a colloquial construction, so I wonder if you would be more likely to find something like ülkeden ülkeye 'from country to country' in written Turkish instead (assuming that is the real meaning of ülke ülke here).
Double words may give different meanings according to the context. 'Ülke ülke' means 'country by country'. You can also say 'şehir şehir'(city), 'ev ev(house)' or more specific things such as 'çanta çanta(bag)'.
And regarding 'güle güle'... It means 'cheerfully, laughing'. And when you say 'güle güle' to someone you want to say goodbye in fact you say 'Go cheerfully. Have no problem on your journey to your destination.' Some other examples:
Ağlaya ağlaya uyudu.(He slept crying./ He cried a lot and slept.)
Yürüye yürüye geleceğim.(I will come on foot/walking.)
Koşa koşa yoruldum.(I got tired of running/through running.)
...-e ...-e suffix indicates procession and the way something is done when it is used with verbs.
"From country to country" is likely the better English translation. "Country by country" isn't a typical expression; it implies that "I am starting at country-A, going to B, then C, then D, etc, and never returning to any, until I've accomplished the task of visiting every country in an area". "From country to country" is common and has a more random connotation: "I'll be travelling throughout the region": there's no implication that I won't reenter a country, nor that I have some kind of "country" bucket-list that I'm trying to check off.
It means "I'm wandering," too, but "I wander" will be marked wrong because it's the simple present tense, and "geziyorum" is the present continuous. I think "I'm wandering country to country" would be a good English translation, though (it sounds more natural to my ear than the existing one), so if that isn't accepted, it should probably be reported.