Je crois que cette haie protège mieux la terrasse du vent que la vieille clôture.

Because haie is a feminine singular noun. The French demonstrative forms are:

• ce for masculine singular before most consonants
• cet for masculine singular before a vowel or mute h
• cette for feminine singular
• ces for plural

So you say cette haie.

It comes from the verb protéger. In the present tense, this verb changes its stem vowel in some forms:

• je protège
• tu protèges
• il/elle protège
• ils/elles protègent

But:

• nous protégeons
• vous protégez

So protège is just the normal third-person singular present form here, agreeing with cette haie.

Because mieux is the comparative form of bien, and it is used as an adverb. Here it modifies the verb protège:

• protège mieux = protects better

Meilleur / meilleure is an adjective, used to describe a noun:

• une meilleure clôture = a better fence

So in this sentence, mieux is the correct choice.

In French, short adverbs such as bien, mal, mieux, souvent, etc. usually go after the conjugated verb in simple tenses.

So:

• protège mieux la terrasse

sounds natural.

Putting mieux later would sound less natural or change the rhythm of the sentence.

Because French commonly uses protéger quelque chose de quelque chose to mean protect something from something.

So:

• protéger la terrasse du vent = protect the terrace from the wind

Here, du is just the contraction of de + le:

• de le vent → du vent

Yes, sometimes contre can be used, especially when the idea is resisting or shielding against something physically. But with protéger, de is extremely common and very natural here:

• protéger la terrasse du vent

So the sentence is idiomatic as written.

Here que means than, not that.

After a comparative like mieux, French uses que:

• mieux ... que = better ... than

So:

• protège mieux ... que la vieille clôture
  means
• protects ... better than the old fence

Yes, French is leaving out repeated words because they are understood from context.

The full idea is:

• This hedge protects the terrace from the wind better than the old fence does.

French often avoids repeating the whole verb phrase when it is obvious. So que la vieille clôture is enough.

Because vieux / vieille usually comes before the noun in French.

So:

• la vieille clôture = the old fence

Also, clôture is feminine, so the adjective must be feminine too:

• masculine: vieux
• feminine: vieille

After affirmative je crois que, French normally uses the indicative, because the speaker is presenting the idea as something believed to be true or likely true.

So:

• Je crois que cette haie protège...

is normal.

The subjunctive is more likely after a negative or a question, for example:

• Je ne crois pas que...
• Crois-tu que... ?