Ich kopiere den Link aus der E‑Mail und füge ihn in die Suchleiste ein.

Because einfügen is a separable verb.

• Infinitive: einfügen = to insert / to paste
• In a normal present-tense main clause, the conjugated part goes to position 2, and the prefix goes to the end:
  • Ich füge ihn in die Suchleiste ein.

This is very common in German:

• ankommen → Ich komme morgen an.
• aufmachen → Ich mache das Fenster auf.

So füge ... ein is just the normal main-clause form of einfügen.

German can leave out the subject in the second clause when it is the same as in the first clause.

So:

• Ich kopiere den Link aus der E-Mail und füge ihn in die Suchleiste ein.

means the same subject is understood in both parts:

• Ich kopiere ... und ich füge ... ein.

Both are grammatical, but leaving out the second ich sounds more natural here.

Because Link is the direct object of kopiere, so it must be in the accusative case.

The noun is masculine:

• nominative: der Link
• accusative: den Link

So:

• der Link = the link, when it is the subject
• den Link = the link, when it is the direct object

Compare:

• Der Link ist lang. = The link is long.
• Ich kopiere den Link. = I copy the link.

Ihn refers back to den Link.

Since Link is masculine singular, the matching pronoun is:

• nominative: er
• accusative: ihn

Here the pronoun is again a direct object of füge ... ein, so accusative is needed:

• Ich füge ihn ein. = I paste it in.

If the noun had a different gender, the pronoun would change:

• das Wort → ich kopiere es
• die Datei → ich öffne sie

Because the preposition aus always takes the dative case.

The noun E-Mail is feminine:

• nominative: die E-Mail
• dative: der E-Mail

So:

• aus der E-Mail = out of / from the email

This is a good pattern to remember:

• aus dem Haus
• aus der Datei
• aus dem Text

Because in can take either accusative or dative, depending on the meaning.

• accusative = movement or direction into something
• dative = location inside something

Here there is a destination idea: you are inserting the link into the search bar. So German uses the accusative:

• in die Suchleiste

Compare:

• Ich füge den Text in die Suchleiste ein. = I put it into the search bar.
• Der Text steht in der Suchleiste. = The text is in the search bar.

They are both first-person singular present tense forms.

• ich kopiere = I copy
• ich füge ein = I insert / paste

The verbs are:

• kopieren
• einfügen

For regular present-tense verbs, ich often takes -e:

• ich mache
• ich lerne
• ich kopiere

And with einfügen, the stem already contains ü:

• fügen → ich füge
• einfügen → ich füge ... ein

Because all nouns are capitalized in German.

So in this sentence:

• Link
• E-Mail
• Suchleiste

are capitalized because they are nouns.

The verbs are not capitalized:

• kopiere
• füge

This is one of the most noticeable spelling differences between German and English.

In this sentence:

• der Link → masculine
• die E-Mail → feminine
• die Suchleiste → feminine

That helps explain the forms you see:

• den Link because masculine accusative
• der E-Mail because feminine dative after aus
• die Suchleiste because feminine accusative after in with direction

Learning nouns together with their article is very important in German.

Yes, in computer language einfügen is the normal German equivalent of paste.

So:

• kopieren = copy
• einfügen = paste / insert

In everyday computer use, Germans often say things like:

• Ich kopiere den Text.
• Ich füge ihn ein.

So even though einfügen literally looks more like insert, it is also the standard word for paste.

Yes, that is grammatically correct.

Adding the second ich can make the sentence sound a little more explicit or slightly more emphatic, but in normal speech and writing, German often leaves it out when the subject is the same.

So both are correct:

• Ich kopiere den Link aus der E-Mail und füge ihn in die Suchleiste ein.
• Ich kopiere den Link aus der E-Mail und ich füge ihn in die Suchleiste ein.

The version without the second ich is usually the more natural one here.