Yes, they do. Only in poetic language the adjective may follow the described thing. Later you will learn that there exists congruence between the adjective and the thing described. This means that they are in the same case: hyvä talo = a good house; hyvässä talossa = in a good house; hyvät talot = good houses. In this respect Finnish differs from the distant Finno-Ugric relatives, like Hungarian, where the adjective remains unchanged before the noun: jó ház = good house; jó házban = in a good house; jó házak = good houses. In this respect, Hungarian follows the original Finno-Ugric pattern, while Finnish has been influenced by the Germanic languages where the form of adjective used to change according to the noun. This is still the case with German (and Icelandic).
We can do this in English as well (although we don't have the case markings), but it's usually either in poetry or in a specific phrase borrowed from another language (usually French) like "attorney general".
How about Estonian? I know it's usually closer to Finnish than Hungarian, but I don't know about this specific detail.
Estonian behaves mainly like Finnish. Here is an example: "a big house - in a big house - big houses" is in Finnish: suuri talo - suuressa talossa - suuret talot, and in Estonian: suur maja - suures majas - suured majad. There are four cases where the case ending is not added to the adjective, Essive ("as a big house") is in Estonian "suure majana", but in Finnish "suurena talona". In Sámi, the adjective remains unchanged with a few exceptions: stuorra dállu - stuorra dálus - stuorra dálut. The Finnish word "maja" = cottage; the Estonian word "talu" = farm. The Sámi word "dállu" means both house and farm. As speakers of Scandinavian languages notice, the Sámi word for big ("stuorra") is borrowed from Scandinavian ("stor" in Swedish and Norwegian).
Finnish and Estonian are so close to each other that it is possible to have a simple everyday conversation - if you know which words to choose and which to avoid. Sámi is definitely further away, and it is not intelligible with Finnish or Estonian. Finnish has had a strong impact on Sámi as the Sámi live in Northern Finland, Sweden and Norway (and in Northern Sweden and Norway the Finnish language used to have a strong position). Thus, if a Finn recognizes a word in Sámi, it is probably a recent loanword from Finnish. Anyway, Sámi is a much closer relative of Finnish than Hungarian is. A layman would not notice anything common between Finnish and Hungarian, although there are several old common words that are still recognizable when pointed out: mikä - mi (what), mitä - mit (what, Partitive / Accusative case), mitkä - mik (which, plural), sarvi - szarv (horn), käsi - kéz (hand), vesi - víz (water), mehiläinen - méh (bee), mennä - menni (to go), niellä - nyel (to swallow), nuolla - nyal (to lick), alla - alatt (under), elävä - eleven (living), and so forth. The first word in the pairs is Finnish, the second is Hungarian. In grammar, there are tens of common features between Finnish and Hungarian.
"Suomalainen" is "Finn" when you are talking about a person (it's a noun), while "suomalainen" is Finnish if you use it as an adjective ("suomalainen ihminen" - a Finnish person, "suomalainen musiikki" - Finnish music, "suomalainen koira" - a Finnish dog). However, "Finnish" can also mean "the Finnish language", but that's translated as "suomi" (lowercase "s") or "suomen kieli".