We adopted an 8-month-old girl from Guatemala in June 2007. She has been quite healthy (the only issue so far is that she needs ear tubes due to fluid in her middle ear), and has not had major adjustment difficulties from moving to a new family and a new country. Guatemala has used a foster care system that in most cases provided excellent care during the wait for the adoption to be finalized. Unfortunately, since January 1 of this year, Guatemala is closed to new international adoptions, as they are in the process of implementing a new law to centralize their adoption system to comply with the Hague treaty. It is not known when they will reopen, or exactly what their new adoption process will be.
Adopting from countries of the former Soviet Union, such as Ukraine and Kazakhstan (or Russia), does bring a higher risk of health problems and emotional adjustment difficulties. This is due to the children living in orphanages rather than foster homes, the fact that they cannot be adopted until they are somewhat older (I believe at least 1 to 1.5 years, maybe older in some countries), and higher prevalence of alcohol use, including by women who are not always very aware of the dangers of drinking while pregnant. I think the majority of these adoptions still turn out well in the end, but you do have to be prepared for a higher risk of problems.
China is also a popular option for adoptive parents (I think the most popular, in fact), because while the children live in orphanages, alcohol use is lower and children can be adopted at a somewhat younger age than in Russia, Ukraine, etc. Also, China's adoption system seems to be more stable and less prone to sudden changes for political reasons (as opposed to Russia or Guatemala, for example). While China did recently tighten their rules for who can qualify as an adoptive parent, these changes were announced well in advance and implemented pretty smoothly.
Regarding costs, definitely that is a major issue, although the tax credit (which I am filing for this year) has now been raised to over $11,000, which certainly helps, at least if you can afford to wait a year or so to get it. Be sure to research your adoption agency carefully before committing to them, check with the Better Business Bureau, etc. We used Commonwealth Adoptions International, which has an office in Cranberry Township, and had a good experience.
Finally, as to your question about nuts - I have traveled to Russia and some other East European countries, although not Ukraine or Kazakhstan specifically. Peanuts do not seem very prevalent in the cuisine of this region. Other nuts are somewhat more so, particularly hazelnuts, it seems. The problem for an allergy situation is that if you are ordering food in a restaurant or buying it in a store, you may not be able to find out exactly what the ingredients are. (Though you may have somewhat better luck buying packaged foods, as the packages are often multilingual and may have an ingredient list in English.) I am not sure about the cuisine in Kazakhstan, because it is in Central Asia rather than Europe, and I expect that the cuisine would show more of an Asian influence.
Sorry this was so long, but I hope it was helpful. Best of luck with your international adoption adventure - it is always an adventure, in one way or another!!