Esta taza no está limpia; dame otra, por favor.

Spanish usually uses estar for a temporary condition or current state, and ser for more permanent characteristics or identity.

So:

• Esta taza no está limpia = This cup is not clean right now.
• Es limpia would sound more like it is a clean one by nature or it is clean in general, which is not what you normally mean here.

For cleanliness, Spanish normally uses estar limpio/a when talking about whether something is clean at the moment.

Because they are two different words:

• esta = this (feminine demonstrative), as in esta taza = this cup
• está = is from the verb estar

The accent mark helps distinguish them in writing.

So in the sentence:

• Esta taza = this cup
• no está limpia = is not clean

Because limpia has to agree with taza, which is a feminine singular noun.

• taza is feminine
• So the adjective must also be feminine: limpia

Compare:

• El vaso está limpio = The glass is clean
• La taza está limpia = The cup is clean

This is standard adjective agreement in Spanish.

Because otra here means another one, and it is being used on its own as a pronoun.

So:

• Dame otra = Give me another one
• Dame otra taza = Give me another cup

Both are correct. In your sentence, taza is understood from the context, so Spanish leaves it out.

Again, it agrees with the implied noun taza, which is feminine.

• otra = feminine
• otro = masculine

So:

• dame otra = give me another one (meaning another cup)
• If the noun were masculine, you would use otro:
  • Este vaso no está limpio; dame otro.

Dame is made of two parts:

• da = give (the tú imperative form of dar)
• me = to me

So:

• da
  • me → dame = give me

In affirmative commands in Spanish, object pronouns are attached to the end of the verb:

• Dime = Tell me
• Ayúdame = Help me
• Tráeme otra taza = Bring me another cup

Because dame is a direct command to tú: give me.

• me da usually means he/she gives me or you give me in the formal usted form, depending on context.
• dame is the informal command used when speaking to one person you address as tú.

Examples:

• Dame otra. = Give me another one.
• Déme otra. = Give me another one. (formal, to usted)

Since this is Spain Spanish, dame is very common in everyday informal speech.

Yes. Por favor is flexible in Spanish.

Your sentence has:

• dame otra, por favor

You could also say:

• por favor, dame otra
• dame, por favor, otra

The most natural options are usually:

• Dame otra, por favor
• Por favor, dame otra

The semicolon links two closely related parts:

• Esta taza no está limpia
• dame otra, por favor

It is similar to saying:

• This cup isn’t clean; give me another one, please.

In normal everyday writing, many people would also use a period or even a comma, depending on style:

• Esta taza no está limpia. Dame otra, por favor.

The semicolon just shows a strong connection between the two ideas.

Spanish distinguishes distance more clearly than English:

• esta taza = this cup (near the speaker)
• esa taza = that cup (near the listener or a bit farther away)
• aquella taza = that cup over there (far from both)

So esta taza suggests the cup is right here, probably the one the speaker is holding or looking at.

Yes, it can be polite enough in the right context, especially with por favor.

• Dame otra, por favor = informal but normal and polite enough with someone you address as tú
• Déme otra, por favor = more formal

Spanish often uses the imperative more naturally than English does, so a direct command is not automatically rude. Adding por favor softens it.

Yes. Spanish often leaves out the noun when it is obvious.

So:

• Esta taza no está limpia; dame otra.

literally means:

• This cup is not clean; give me another.

But in natural English, you would usually say:

• give me another one
• or give me another cup

Spanish does not need an extra word like one here.

Yes, it sounds natural. A speaker in Spain might say it in a café, restaurant, office, or home.

Very natural alternatives include:

• Esta taza no está limpia. Dame otra, por favor.
• Esta taza está sucia; dame otra, por favor.
  (sucia = dirty)

The original sentence is perfectly normal and understandable.