What is the difference between annexing, occupying and taking over a country?

The Annexation of Austria was done to supposedly bring order in a neighbouring country where there was political chaos making use of a vacuum in power. By annexing the annexed territory becomes part of the annexing nation. Occupation brings in an army but can use puppet politicians from within the country like the Vichy regime in France. A take over is a more colloquial term referring to the taking over of political and/or military power and has to be specified more in the context used. If you look at Sudetenland it was annexed from Czechoslovakia and eventually the rest of Czechoslovakia was occupied.

Military occupation of enemy territory in war is a temporary war measure which is usually ended by the peace treaty in a few months or years.

Annexation, often the result of peace treaties at the ends of wars, is intended to be permanent.

Of course, because nothing is certain in politics, war, and diplomacy, some "temporary" occupations have last longer than some "permanent" annexations, but as a general rule annexations last many times longer than occupations, often for centuries.

Here is a link to a map of German territory in World War II.

https://en.wikipedia.org/wiki/Nazi_Germany#/media/File:German_Reich_1942.png1

The darkest area, the German Reich, contains the original German Reich and territories already annexed by Hitler, intended to be permanently German. The lighter colored occupied regions include areas Hitler may or may not have intended to stop occupying at the end of the war and also territories in the East that Hitler intended to annex to Germany at the end of the war or before, but had not yet done so.

Question:
What is the difference between annexing, occupying and taking over a country?

Annexing is when you add a territory to your country. Where you say this is ours now, it's within our boarders, falls under our laws. An example would be Israel annexation of the Golan Heights. It's typically a designation which occurs after "taking over" and "occupation". Golan Heights taken by Israel in 1967, then occupied for 14 years, finally annexed by Israel in Dec 1981.

Occupation is typically generally temporary designation, what occurs after a war to ensure no future war. The allies occupied Germany and Japan after WWII. It doesn't necessarily change borders or national designations. It's the positioning of foreign troops inside a country, and placing that country under your governance, generally military governance. It differs from annexation in that the occupier acknowledges the land is still not their possession even while they station troops there, provide for it's security and governance. Occupation generally follows conquest or taking over and sometimes occupation turns into annexation.

Taking over a country is probable the most arbitrary case. It could be a revolution where an internal force "took over" the country. Like the Communists took over Russia in 1917, or China is 1949. It's not an annexation because to merely take over a country or region doesn't speak to it's future designation and relationship as an independent country / region. "Taking over" in it's basic forms just means a change of who's in control without speaking to why or any future plans.

Question: When Hitler took a sliver of Poland, and Czechoslovakia, did he take over or did he occupy it?

Hitler did annex parts of Poland and Czechoslovakia in WWII. Further he took over and occupied large swaths of the countries. All of Czechoslovakia and about half Poland which he split with his then ally the Soviet Union.

Hitler set up independent states in both countries, which were puppet governments answerable and dependent upon Germany. If Germany had won the war no telling what their future arrangement would have been, but it's pretty clear it would have been favored the interests and desires of Germany.