A fita adesiva está na caixa das ferramentas, ao lado da chave de fendas.

Because the sentence is talking about location: where the tape is.

In Portuguese, estar is normally used for:

• where something or someone is
• temporary states
• conditions

So A fita adesiva está na caixa... means the tape is located in the box.

By contrast, ser (as in é) is used for:

• identity
• permanent or defining characteristics
• time, origin, profession, etc.

So:

• A fita adesiva está na caixa. = The tape is in the box.
• Isto é fita adesiva. = This is adhesive tape.

Fita adesiva means adhesive tape or sticky tape.

A few useful notes:

• fita = tape/ribbon/strip
• adesiva = adhesive, sticky

In everyday European Portuguese, people may also say fita-cola, which is very common in casual speech for sticky tape.
Fita adesiva is perfectly correct and a bit more neutral or descriptive.

A is the definite article the, and it agrees with fita, which is a feminine singular noun.

So:

• a fita = the tape
• o martelo = the hammer
• as ferramentas = the tools
• os parafusos = the screws

Portuguese uses definite articles very often, sometimes more often than English does. In this sentence, a fita adesiva refers to a specific piece or roll of tape.

They are contractions, which are extremely common in Portuguese.

Here is how they work:

• na = em + a
  na caixa = in the box

• das = de + as
  das ferramentas = of the tools

• da = de + a
  da chave = of the screwdriver

These contractions are standard and normally required. Portuguese uses them much more systematically than English uses similar combinations.

Yes, caixa de ferramentas is also possible, and in many contexts it is actually the more idiomatic way to say toolbox.

The difference is roughly this:

• caixa de ferramentas = toolbox, tool box in a general sense
• caixa das ferramentas = the box of the tools, the box where the tools are

So in this sentence, caixa das ferramentas sounds like a box associated with a specific set of tools already known in the context.
If you simply mean the toolbox, many speakers would naturally say a caixa de ferramentas.

Because ferramentas means tools, plural.

A toolbox is associated with multiple tools, so Portuguese commonly uses the plural noun here:

• ferramenta = tool
• ferramentas = tools

So caixa de ferramentas or caixa das ferramentas refers to a box for tools, not just one tool.

Ao lado de is a fixed expression meaning next to, beside, or at the side of.

In the sentence, it appears as ao lado da chave de fendas.

Breakdown:

• ao = a + o
• lado = side
• de = of
• da = de + a

So:

• ao lado de = next to
• ao lado da chave de fendas = next to the screwdriver

Even though English says next to the screwdriver, Portuguese uses this longer structure.

Yes. Chave de fendas is the standard European Portuguese term for a screwdriver, especially the traditional flat-head type.

Word by word:

• chave = key / tool / wrench-like instrument
• fendas = slots, slits

So historically it is something like slotted-screw tool.

This is a very Portuguese way of naming tools: chave is used in many tool names.

In European Portuguese:

• chave de fendas = screwdriver
• chave Phillips = Phillips screwdriver

Because in Portuguese, adjectives often come after the noun.

So:

• fita adesiva = adhesive tape
• caixa grande = big box
• chave pequena = small screwdriver

Putting the adjective after the noun is the normal pattern here. Some adjectives can go before the noun, but that often changes the tone or meaning slightly. With adesiva, the usual order is definitely fita adesiva.

Yes, Portuguese word order is somewhat flexible, especially with location phrases.

The original sentence is completely natural:

• A fita adesiva está na caixa das ferramentas, ao lado da chave de fendas.

But you could also say:

• A fita adesiva está ao lado da chave de fendas, na caixa das ferramentas.
• Na caixa das ferramentas, ao lado da chave de fendas, está a fita adesiva.

The meaning stays basically the same, but the focus changes:

• starting with A fita adesiva focuses on the object
• starting with Na caixa das ferramentas focuses on the location

Because Portuguese commonly uses definite articles where English may use fewer.

In this sentence you get:

• a fita adesiva
• na caixa
• das ferramentas
• da chave de fendas

This is very normal. Portuguese often marks nouns as definite when the speaker has a specific item in mind. English sometimes sounds more natural with fewer articles, but Portuguese tends to be more explicit.

So even if it feels article-heavy to an English speaker, this is standard Portuguese structure.