Hi Andrew, your example of (the?) Ukraine is in interesting one, and the current convention is to avoid "the". For most other cases, two-word (or more) countries tend to be descriptive: the United Kingdom (not just any kingdom), the Czech Republic (as opposed to the other republics - I don't think many people know we can say "Czechia", by the way) and so on.
Geographical areas sometimes remove the noun and just use the adjective. For example, "the Alps" = the Alpine mountains, and "the Philippines" actually means the Philippine islands, ie. the islands of King Philip. Dordogne, Pas de Calais and Auvergne all suggest "the ____ region".
These are just general guides. Sometimes we keep an article because it translates literally, eg. The Hague = Den Haag. Perhaps it's simplest to just learn each one as you go.