Yes, these are correct and perfectly natural. However, this is not to possessive sense of あります, but the existential sense. Refer to aru Kanji confusion for additional explanation.
先週の金曜日に試験がありました。 → Last (week on) Friday there was/existed a test.
And contrary to @Zeyuan's answer, you do not need to use ~には. You can use either of ~に or ~には. The former is simply neutral while the latter puts emphasis on the time.
As far as the ~は~です pattern, remember that the particles and copula are as flexible as they are in English. 試験は金曜日です is fine to say "The exam is Friday", and just as in English, it doesn't mean "The test equals Friday" or that it literally is Friday as you say.
- [幸子]{さち・こ}: [雅人]{まさ・と}君どこ出身ですか? → Masato, where is your hometown?
- 雅人: 名古屋です。幸子さんは? → I'm from Nagoya. What about you, Sachiko?
- 幸子: 私は大阪です。 →
I am Osaka (As for me) It's Osaka
You wouldn't equate Sachiko to literally being the same thing as Osaka. Remember that は is "simply" the topic marker.