Learn Portuguese (Portugal): Tive uma gripe grave.

Tive is the simple past tense (Pretérito Perfeito) of the irregular verb ter (to have) for the eu (I) form. You use this tense here because having the flu was a specific, completed event in the past.

Why isn't the word eu (I) used at the beginning of the sentence?

In Portuguese, it is very common to drop subject pronouns like eu. Because the verb form tive uniquely means I had, adding eu is not necessary unless you want to add special emphasis.

Since gripe is a feminine noun, why doesn't the adjective change to grava?

Adjectives that end in -e in Portuguese, such as grave, triste, or grande, are invariable for gender. This means they keep the exact same form whether the noun they describe is masculine or feminine. They only change when made plural by adding an -s.

Does grave mean the same as the English word grave?

They are related, but in Portuguese, grave is an adjective that translates to severe or serious, especially when talking about an illness, an injury, or a problem. If you mean a grave in a cemetery, you would use nouns like túmulo or sepultura.