“还是” is a phrase that suggesting there's a choice to be made. so it should be placed between two items. By “item”, it can be all kinds of words, either noun “中国，俄国”, or verb, or even a word-combination, such as "喝茶" “喝啤酒”. e.g. —“她和我分手了，我不想活了” — “那你是想哭，还是想闹，还是想上吊？” items are verbs. Are you with me now? 你（选择）喝茶还是喝啤酒。 "选择" is left out as a verb.

Michael, If you really must translate (我不鼓励。我赞成学中文要全盘用中文，一句英语也不用。That is how the Alliance Française teaches French - purely in French.),  你喝茶还是喝啤酒 is likely to be one of these two 1. Are you going to have tea or beer? 2. Would you like some tea or beer? Your "before the verb or after the verb" analysis would not help you best. Use the concept of "A还是B" instead. A and B are the options. They can be nouns (including gerunds) or verbs (including the infinitif). I don't advocate, and in fact would advise against, however, learning Chinese using English grammar. A还是B - examples showing different applications 美国大还是英国大? 你爱爸爸还是爱妈妈? 你喜欢男孩还是女孩? 你要离婚还是强忍你老婆偷汉子? 继续抵抗还是马上投降?

Michael, The real verb in 你觉得中国大还是俄国大 is "大"! "中国大" translates as "China is big(ger)". The complete sentence in English would be, "Do you think China is bigger or Russia is bigger?" So the structure "A还是B" is in fact similarly used in both your sentences. Do not be misled by the "Do you think" (你觉得)。

in Chinese , 喝茶 is a short sentence, e.g. 你想喝点什么？ 喝茶。 what would you like to drink? A cup of tea. 中国大 is also a short sentence, 除了俄国，亚洲哪个国家大？中国大. what is the beggest contry in Asia except Russion？China. so you can not split 喝 and 茶，中国 and 大，bear or tea, that is English sentence custom, 喝茶还是喝啤酒 or 茶还是啤酒 is Chinese sentence custom , of cause, 喝茶还是啤酒 is also right in Chinese

你喝茶还是喝啤酒？你觉得中国大还是俄国大。 Both are correct. In chinese 还是 could be a conjunction to use in sentense 还是觉得。I prefer to say that you can think its a verb by combining 4 words to express the subjective feelings of your. For example: 我还是认为你是对的-I am still thinking that you are right 我还是认为外面可能在下雪。-I am still thinking that there is maybe too cold outside.