هو يقترح نزهة.

Because the verb يقترح already contains the subject he inside its form: it is 3rd person masculine singular.

So:

• يقترح نزهةً = He suggests an outing
• هو يقترح نزهةً = also He suggests an outing

Adding هو can give extra emphasis, make the subject clearer, or simply reflect a teaching style that shows the subject explicitly.

يقترح is the present/imperfect tense form of اقترح.

It means:

• he suggests
• he is suggesting
• sometimes he proposes

So in context, Arabic present/imperfect can cover both simple present and present progressive, depending on the situation.

Arabic does not have a separate indefinite article like a/an.

Instead, a noun is usually indefinite when:

• it does not have ال
• and, in fully vowelled text, it often takes tanwīn

So:

• النزهة = the outing
• نزهة / نزهةً = an outing

That is how Arabic expresses the idea of a here.

In fully correct, fully vowelled MSA, yes: نزهةً is the expected form here.

Why?

Because نزهة is the direct object of يقترح, so it should be in the accusative case.

So the fully marked version is:

• هو يقترح نزهةً

But in normal everyday Arabic writing, short vowels and case endings are usually omitted, so writing نزهة is completely normal.

Yes. In this sentence, نزهة is what he is suggesting, so it is the direct object.

Structure:

• هو = subject pronoun
• يقترح = verb
• نزهةً = direct object

So the sentence is basically:

• He
  • suggests
    • an outing

Yes, absolutely.

In fact, Arabic often prefers to leave out subject pronouns when the verb already makes the subject clear.

So both are possible:

• يقترح نزهةً
• هو يقترح نزهةً

The version without هو is often more natural unless you want to emphasize he.

Not if what follows is a noun, as in this sentence.

So this is fine:

• يقترح نزهةً = He suggests an outing

But if he is suggesting an action, Arabic often uses another structure, for example:

• يقترح أن نخرج = He suggests that we go out
• يقترح الذهابَ في نزهةٍ = He suggests going on an outing
• يقترح القيامَ بنزهةٍ = He suggests taking an outing / going for a walk

So يقترح can be followed by either a noun or a verbal idea, depending on what you want to say.

A careful full pronunciation would be approximately:

• huwa yaqtariḥu nuzhatan

A few notes:

• يقترح = yaq-ta-ri-ḥu
• نزهةً = nuz-ha-tan

The final ة is a tāʾ marbūṭa.
In pause, learners often hear it more like -ah, but with full case endings it is pronounced with t before the ending: nuzhatan.

It is understandable and grammatically useful, especially for learning.

But depending on the exact meaning in English, Arabic may often use a slightly fuller expression, such as:

• يقترح القيامَ بنزهةٍ
• يقترح الذهابَ في نزهةٍ

These can sound more natural if the idea is specifically he suggests going for a walk / going on an outing.

So هو يقترح نزهةً is fine as a simple learning sentence, but in real usage Arabic may prefer a more explicit phrasing.