Learn French: Le président dit que la crise est grave. - The president says that the crisis is serious.

In English, we often drop "that" and just say "The president says the crisis is serious." In French, you can never drop the conjunction que. It must always connect the introductory verb (dit) to the reported statement.

Since this is reported speech, do I need to change the tense of the second verb (est)?

Not in this case. When the introductory verb is in the present tense (like dit here), the tense of the original quote doesn't change. The president's original words were "La crise est grave" (present tense), so it stays in the present tense when reported.

Why isn't the subjunctive used after que in this sentence?

Verbs that declare a fact or state an opinion, like dire (to say), take the normal indicative mood when they are affirmative. You generally only trigger the subjunctive when expressing doubt, desire, emotion, or necessity.

Does grave have a special feminine spelling to match la crise?

It is feminine here, but it happens to look the same! Adjectives must agree with the noun they describe. However, because the masculine form of grave already ends in an "e", you don't add another one for the feminine form.