Learn Finnish: Ainoa ongelma on hinta. - The only problem is the price.

In an A equals B sentence where both parts are specific, countable nouns, Finnish uses the nominative case for both. You are equating ainoa ongelma with hinta. If the second word were an adjective describing the problem, it might take the partitive, but since they are both nouns representing a specific identity, they stay nominative.

How would this sentence change if I used toinen instead of ainoa, given this lesson's theme?

Swapping the word changes the reference entirely. Toinen ongelma on hinta means the other problem is the price, implying there is at least one more problem. Ainoa is used when there are zero other options or items, meaning the only one.

Does the word ainoa behave like a normal adjective when the sentence requires a different case?

Yes, ainoa modifies the noun it is attached to and must agree with it in case and number. For example, if the grammar required the partitive case, you would use ainoaa ongelmaa. Here, both words stay nominative because they are the subject.

Could I reverse the word order and say Hinta on ainoa ongelma?

Absolutely. Hinta on ainoa ongelma translates to the price is the only problem. Because Finnish uses grammatical cases rather than strict word order to show how words relate, you can safely flip these sentences to change the emphasis. Putting ainoa ongelma first highlights right away that there is only a single issue.