Veo el cartel borroso.

Because Spanish often leaves out subject pronouns when they are clear from the verb ending.

• veo already means I see
• the -o ending tells you it is first person singular

So Veo el cartel borroso is a completely natural sentence.

You can add yo if you want emphasis or contrast:

• Yo veo el cartel borroso, pero tú no. = I see the sign blurry, but you don’t.

Usually no. In Spain, cartel most commonly means something like:

• poster
• sign
• notice
• sometimes billboard, depending on context

So in this sentence it refers to something you can read or look at.

The English meaning cartel as a criminal or business cartel is usually cártel in Spanish, with an accent mark.

Because cartel is a masculine singular noun, so it takes el in the singular:

• el cartel
• los carteles

Also, el shows that we are talking about a specific sign/poster, not just any one.

Compare:

• Veo el cartel. = I see the sign/poster.
• Veo un cartel. = I see a sign/poster.

Because this is a direct object, and Spanish only uses the personal a mainly with people, and sometimes animals or personified beings.

So:

• Veo el cartel. = correct
• Veo a Juan. = correct
• Veo a el cartel. = incorrect

Since cartel is an inanimate thing, there is no personal a.

Here borroso comes after the noun because it is describing the object as it is perceived.

A very common Spanish pattern is:

verb of perception + object + adjective

For example:

• Veo el cartel borroso.
• Oigo la música alta.
• Encontré la puerta abierta.

In this kind of structure, the adjective tells you the state or quality of the object as the speaker sees, hears, or finds it.

Because borroso is an adjective, and here it describes el cartel.

• borroso = blurry
• borrosamente = blurily / in a blurry way

In this sentence, Spanish is not saying that the action of seeing happens blurily as an adverb. It is saying that the sign appears blurry to the speaker.

So:

• Veo el cartel borroso. = natural
• Veo el cartel borrosamente. = not the normal way to say this

Because adjectives must agree with the noun they describe.

Here, cartel is:

• masculine
• singular

So the adjective must also be masculine singular:

• el cartel borroso

Compare:

• la señal borrosa
• los carteles borrosos
• las señales borrosas

Because ver and mirar are not the same.

• ver = to see
• mirar = to look at

Veo el cartel borroso means the speaker’s visual perception is blurry.

If you say miro el cartel, that means you are directing your eyes toward it, but it does not by itself say how clearly you see it.

A useful contrast:

• Miro el cartel, pero lo veo borroso.
  = I’m looking at the sign, but I see it blurry.

In this sentence, the most natural interpretation is that it describes how the sign appears to me.

So Veo el cartel borroso usually means something like:

• I see the sign blurry.
• The sign looks blurry to me.

This is slightly different from simply identifying it as the blurry sign.

If you want to make the idea even clearer, Spanish can also say:

• Veo borroso el cartel.
• Veo el cartel, pero lo veo borroso.

Context usually makes the intended meaning clear.

Yes, Spanish has some flexibility.

You may also hear:

• Veo borroso el cartel.

This is also possible, and it can put a bit more focus on borroso.

But Veo el cartel borroso is a normal and clear sentence. For a learner, it is a good standard pattern to remember:

ver + object + adjective