Learn Russian: Интервал между поездами — тридцать секунд. - The interval between the trains is thirty seconds.

In Russian, the present tense of the verb "to be" is normally skipped. When you link a noun subject to another noun or a numeral (like saying "the interval [is] thirty"), an em-dash is used in writing to take the place of "is".

Why is поездами in the instrumental case here?

It follows the preposition между (between). The preposition между requires the noun that follows it to be in the instrumental case. For the plural noun поезда (trains), the instrumental form is поездами.

Why is the word for "seconds" (секунд) in the genitive plural?

When counting with numbers ending in 0, or 5 through 9 (like тридцать), the following noun must be in the genitive plural. Since секунда is a feminine noun ending in -а, its genitive plural form drops the final vowel, leaving just секунд.

Since this lesson covers reading the clock, why don't we use a genitive ordinal number here, like we do for "five minutes past one"?

The structure using ordinal numbers (like пять минут первого) is exclusively used for telling a specific time of day on the clock. This sentence is simply expressing a duration or quantity of time (an interval of thirty seconds), so we just use standard counting numbers.