Jeg bruger søgefeltet på hjemmesiden, men jeg har for mange faneblade åbne, så computeren bliver langsom.

Those endings are the Danish way of saying the.

• et søgefelt = a search field

• søgefeltet = the search field

• en hjemmeside = a website/homepage

• hjemmesiden = the website/homepage

• en computer = a computer
• computeren = the computer

In Danish, the definite article is usually attached to the noun instead of being a separate word as in English.

Yes. Danish very often makes compound nouns as a single word.

Here:

• søge = search
• felt = field
• søgefelt = search field
• søgefeltet = the search field

This is very normal in Danish. English often writes these as two words, but Danish usually joins them.

Because Danish normally says på en hjemmeside for on a website.

So:

• på hjemmesiden = on the website

This works like English on the website, on the page, on the internet.
Using i would usually sound less natural here, because websites are thought of as surfaces or platforms you are on, not containers you are in.

Because men joins two full clauses, and the second clause needs its own subject.

So the sentence is:

• Jeg bruger søgefeltet på hjemmesiden
• men jeg har for mange faneblade åbne

Danish usually states the subject again in the second clause, just as English does in I use..., but I have...

Yes. In Danish, for in this kind of phrase means too.

• for mange = too many
• for meget = too much
• for få = too few

So jeg har for mange faneblade åbne means I have too many tabs open, not for many tabs.

Yes. Faneblad is the standard Danish word for a tab, especially in computers and browsers.

Forms:

• et faneblad = a tab
• faneblade = tabs
• fanebladet = the tab
• fanebladene = the tabs

So for mange faneblade means too many tabs.

Here åbne is an adjective, not a verb.

The structure is:

• jeg har
  • object + adjective

So:

• jeg har vinduet åbent = I have the window open
• jeg har døren lukket = I have the door closed
• jeg har for mange faneblade åbne = I have too many tabs open

This is a very common Danish pattern. The adjective describes the state of the object.

Also, åbne has -e because it agrees with the plural noun faneblade.

Yes, you could. Both are possible, but they are slightly different in structure.

• for mange åbne faneblade = too many open tabs
• for mange faneblade åbne = have too many tabs open

The version in your sentence is especially natural when talking about the current state of something you have open, closed, ready, finished, and so on.

Because bliver means becomes or gets.

• computeren bliver langsom = the computer becomes slow / gets slow
• computeren er langsom = the computer is slow

In this sentence, the many open tabs cause the computer to change state, so bliver fits better.

Because adjective endings in Danish depend on what they describe.

Here computeren is a common-gender singular noun, so the adjective is:

• langsom

Compare:

• en computer er langsom
• et program er langsomt
• to computere er langsomme

So in your sentence, computeren bliver langsom is exactly what we expect.

Here så means so, introducing a result:

• ..., så computeren bliver langsom = ..., so the computer becomes slow

In this use, så functions like a conjunction linking clauses. The clause itself starts with computeren, so the normal order is:

• subject + verb
• computeren bliver

If you used a fronted adverb like derfor meaning therefore, then you would get the usual Danish main-clause inversion:

• Derfor bliver computeren langsom

So the word order in your sentence is normal.

It can mean either, depending on context.

Danish present tense often covers both:

• I use
• I am using

So jeg bruger søgefeltet på hjemmesiden could mean:

• I use the search field on the website
• I am using the search field on the website

The surrounding context tells you which meaning is intended.