Я заклею коробку скотчем перед переездом.

Заклею is the 1st person singular future form of заклеить, a perfective verb.

The base idea is клеить = to glue / stick / tape.
With the prefix за-, заклеить often means to seal up, to tape shut, or to cover by sticking something over it.

So in this sentence, заклею коробку means not just I will tape the box, but more specifically I will tape it shut / seal it with tape.

Russian perfective verbs do not have a true present tense. Their present-looking forms usually express the simple future.

So:

• я клею = I am taping / I tape (imperfective, present)
• я заклею = I will tape shut / I will seal (perfective, future)

Because заклеить is perfective, заклею naturally means I will tape it shut.

Because коробка is the direct object of the verb, it goes into the accusative case.

• Nominative: коробка = box
• Accusative: коробку = box as the thing being acted on

Since the speaker is doing something to the box, Russian uses the accusative:

• Я заклею коробку = I will tape the box shut

This is a very common pattern with transitive verbs.

Скотчем is instrumental because it expresses the means/tool/material used to do the action.

Compare:

• писать ручкой = to write with a pen
• резать ножом = to cut with a knife
• заклеить скотчем = to seal with tape

So скотчем means with tape.

Its dictionary form is скотч, and the instrumental singular form is скотчем.

Yes, in Russian скотч very commonly means Scotch tape / adhesive tape in everyday speech.

So:

• скотч = tape
• скотчем = with tape

Context usually makes the meaning obvious. In a sentence about sealing a box, it clearly means tape, not whisky.

The preposition перед means before in this sentence, and it requires the instrumental case.

So:

• dictionary form: переезд = move / relocation
• instrumental form: переездом

That gives:

• перед переездом = before the move

This is a fixed grammar pattern worth remembering:

• перед + instrumental = before
• перед домом = in front of the house / literally before the house
• перед встречей = before the meeting
• перед переездом = before the move

Here переезд means a move, as in moving house or relocating.

So:

• перед переездом = before moving / before the move

Be aware that переезд can also mean other kinds of crossing or moving in different contexts, but in everyday home-related situations it often means relocation.

Yes, it could be omitted.

Russian often drops subject pronouns when the verb form already makes the subject clear.

• Заклею коробку скотчем перед переездом.
  = I’ll tape the box shut before the move.

Because заклею clearly means I will, the pronoun я is not strictly necessary.

Including Я can add emphasis, contrast, or simply sound a little more explicit.

Yes. Russian word order is flexible, though different orders can shift emphasis.

Neutral order here is:

• Я заклею коробку скотчем перед переездом.

Other possible versions:

• Перед переездом я заклею коробку скотчем.
  Emphasizes before the move.

• Коробку я заклею скотчем перед переездом.
  Emphasizes the box.

• Скотчем я заклею коробку перед переездом.
  Emphasizes with tape.

The basic meaning stays the same, but the focus changes.

You could use клеить, but it would not mean quite the same thing.

• Я клею коробку скотчем sounds less natural for sealing a box shut
• Я заклею коробку скотчем clearly means I will tape the box shut / seal it with tape

So заклеить is the better choice when the action is completed and the result is that the box is sealed.

Заклею is pronounced roughly like za-KLYE-yu.

It comes from заклеить. In the future/“present” forms of this perfective verb, the ending changes:

• я заклею
• ты заклеишь
• он/она заклеит

So the я form is simply заклею, not something longer like заклеию. This is just the normal conjugation pattern of verbs like клеить / заклеить.