Position of a relative clause before/after a verb

There are two ways of saying this (given only what you provide in your question):

All kinds of problems that smaller animals or plants do not have to cope with arise. (This sentence, however, separates the verb arise from the subject of the sentence.)

There arise all kinds of problems that smaller animals or plants do not have to cope with. (This sentence, however, is much more formal and perhaps stilted. It's strictly written English.)

Perhaps it would be better to start off with the reason those problems arise, e.g.:

When X happens, it causes all kinds of problems that smaller animals or plants do not have to cope with.

or

X causes all kinds of problems that smaller animals or plants do not have to cope with.

There's a well-known syntactic rule called Extraposition from Noun Phrase that takes a heavy relative clause modifying an argument NP and "moves" it to the end of a sentence, which is the preferred place for heavy material in English, since it's a right-branching language.

Extraposition does the same thing, except it "moves" a subject complement clause and leaves a dummy it subject behind. Extraposition from NP leaves no dummy, and just expects the listener to pick up the relative clause after the verb again; so it's less common and much more prevalent with extremely short verbs, as here; the longer the verb is, the less likely the listener is to be ready to back up and parse the subject again.

Here's the input sentence, which I think we will all agree is awkward at best.
(The verb phrase is boldfaced below, and the relative clause is [bracketed])

• All kinds of problems [that smaller animals or plants do not have to cope with] arise.

Extraposition from NP changes this to the sentence in the question

• All kinds of problems arise [that smaller animals or plants do not have to cope with].

This also has the effect of putting the verb up front, instead of at the end. English prefers to have the verb as the second constituent in the sentence, right after the subject NP. Variations from that prototype order are used to mark special phenomena, like questions and relative clauses.