The meaning is that this (thing) will be / exist. For how long? Forever. This (thing) will be (will exist) forever / for all eternity / always. You could end the sentence at 'be'. Or you could use any how long info.

I think "this will be forever" sounds a bit weird too, so the question might have a flawed premise. What is the context? In both sentences, if the last word is an adverb, then the word it's modifying is "be." To be is to exist. Saying that something will exist forever is pretty normal (especially as hyperbole). Saying "the way this thing is going to exist is unexpectedly" is very weird. I guess it's technically a grammatically correct sentence, but not something that anyone would ever say. Usually when we say "this will be x," the "x" is almost always an adjective modifying "this," not an adverb modifying "be." This is the much more common and natural way of speaking. So you would say "this will be unexpected." With "this will be unexpectedly," I find myself waiting for the adjective. It will be unexpectedly what? Unexpectedly fast? Unexpectedly funny? I hope this helps!

It is not incorrect, but the sentence would only be useful in a context where the missing adjective is understood. The sentence cries out for a predicative complement, such as "This will be unexpectedly FUN." However, if spoken in a context where everyone knows you are talking about "fun", the sentence is good.

"Forever" has multiple slightly different meanings. In your sentence, it is used as an adjective, and means "permanent". In other sentences, it can be used as an adverb, and means "permanently". Similarly, you can say "this will be unexpected". Now this sentence is using an adjective, and is correct.