هذه مشكلة جادة.

Yes. مشكلة is grammatically feminine. One common clue is the ending ة (tāʾ marbūṭa), which often marks feminine nouns in Arabic.

Also, grammatical gender in Arabic is not about biological sex only. Even an inanimate thing like a problem can be feminine.

Because هذه is the demonstrative this for a feminine singular noun. Since مشكلة is feminine singular, the correct form is هذه.

If the noun were masculine, you would usually use هذا instead.

In Arabic, the present tense of to be is usually not stated in this kind of sentence. So Arabic often says literally something like this serious problem, while the meaning is This is a serious problem.

If you wanted the past tense, you would use a verb such as كان.

Here it is a full sentence. هذه means this, and مشكلة جادة is the description/predicate: a serious problem.

That is different from هذه المشكلة الجادة, which would normally mean this serious problem as a noun phrase.

Because the sentence is identifying or classifying something: This is a serious problem. In Arabic, the predicate in this kind of sentence is often indefinite.

If you said هذه المشكلة..., you would be talking about the problem, a specific known one.

Because in Arabic, adjectives normally come after the noun they describe. So مشكلة جادة is literally problem serious, but it means serious problem.

This is the normal word order for noun + adjective in Arabic.

Because Arabic adjectives must agree with the nouns they describe. Since مشكلة is feminine singular, the adjective must also be feminine singular: جادة.

The masculine form is جادّ, and the feminine form is جادّة.

Because adjectives in Arabic also agree in definiteness. Since مشكلة is indefinite, جادة must be indefinite too.

So:

• مشكلة جادة = a serious problem
• المشكلة الجادة = the serious problem

When you stop at the word, the final ة is usually pronounced like -a or -ah. So you will often hear mushkila and jādda.

In fully pronounced formal Arabic with case endings, it can sound like -tun: mushkilatun, jāddatun.

A careful, fully pronounced reading is:

hādhihi mushkilatun jāddatun

In normal pause, many speakers would say something closer to:

hādhihi mushkila jādda

The dh in hādhihi sounds like th in this, not like th in thin.

Because normal Arabic writing usually leaves short-vowel marks out. Learners often see them in textbooks, but everyday Arabic text usually does not include them.

A fully vocalized version would be:

هٰذِهِ مُشْكِلَةٌ جَادَّةٌ

Those marks help show pronunciation and grammar, but native readers usually read the unvocalized form easily from context.