This answer applies to U.S. English. The meanings overlap. One of the dictionary meanings of the noun "alternate" is "alternative." Your four examples are all fine, but you could either word in any of the four sentences and nobody could say that it was wrong. In the United States, "alternate routes" is a stock phrase used on road signs and traffic reports, so for me, as a U.S. speaker, it is the more natural choice for #2. "Alternative energy" or is a stock phrase (as is "alternative power" and "alternative investments") and therefore you have picked the more natural choice for #3. "Alternative" sometimes carries the idea of something less usual. "Alternative" also carries the idea of a forced choice between two things. "Leaving me no alternative but..." and "having no alternative but" are stock phrase where "alternate" cannot be used. "We needed a loaf of bread on New Year's Day and the supermarket was closed, so I had no alternative but to use the little convenience store."

Your two examples with 'alternate' don't seem right to me. I would never use 'alternate' in that way. 'Alternative' (referring to another option which is available) would be more appropriate in both those cases: alternative plans and alternative route. The adjective 'alternate' (stress on the second syllable) means that you alternate (verb - stress on the first syllable) between one thing and another. 'My co-worker and I work alternate days. I do Monday and Wednesdays, for example, and she does Tuesdays and Thursdays'. I believe you can use 'alternate' in N.American English in the first sense (as in 'an alternate reality'). However, if the purpose of your exercise is to distinguish between these two different adjectives, that's what you need to do - distinguish between them.

It depends if you are following the rules of American or British English (I am British). To me, alternate means to take something in turns, for example, we work alternate fridays...i work on one friday, my colleague works the alternate friday. Alternative means a different option, so I would say....'for rainy days make sure you have alternative plans' NOT alternate. But Americans would probably say 'alternate plans'. Same with 2. I would say 'we have to take an alternative route'.

'Alternate' (AmE) is something that you can do or can use if your 'first choice' fails. It's your 'reserve thing' to reach your goal. 'Alternative' is something that you can VOLUNTARILY choose if you think it's a different thing or solution from, and better than, another. Sic et simpliciter. (Of course in AmE 'alternate' also has the meaning of 'every other' (for instance 'alternate days' = 'every other day') and other similar meanings, like in BrE).

should be OK