Learn Russian: Здесь тихо и спокойно.

In Russian, when describing the general state of an environment, atmosphere, or weather without a specific subject (like the dummy 'it' in English), we use impersonal constructions. These require the predicative adverb form—which looks exactly like the short neuter adjective—because there is no specific noun for a regular adjective to agree with.

Since there is no verb in this sentence, how would we change it to the past or future tense?

This sentence uses a 'zero copula' (an omitted verb 'is') for the present tense. To talk about the past or future, you just add the neuter singular form of the verb to be: было for the past (Здесь было тихо и спокойно) and будет for the future (Здесь будет тихо и спокойно).

Does the word order matter here? Could I say Тихо и спокойно здесь?

You can, but it changes the nuance. Russian word order is flexible but uses the end of the sentence to highlight new or important information. Здесь тихо и спокойно emphasizes the state of the place (it is quiet and calm). Reversing it to Тихо и спокойно здесь places the emphasis on the location, as if you were specifically answering the question 'Where is it quiet and calm?'