Meine Schwester nimmt eine Stecknadel und steckt das Papier fest.

Because Schwester is a feminine noun, and mein- has to take the correct ending.

Here, meine Schwester is the subject of the sentence, so it is in the nominative case.
For a feminine noun in the nominative, mein- becomes meine.

Compare:

• mein Bruder = my brother
• meine Schwester = my sister
• mein Kind = my child

So meine Schwester is the correct form for my sister.

Both are in the accusative case because they are the direct objects of the verbs:

• nimmt eine Stecknadel → she takes a pin
• steckt das Papier fest → she fastens the paper

A quick breakdown:

• eine Stecknadel: feminine singular accusative
  
  • for feminine nouns, eine is used in both nominative and accusative
• das Papier: neuter singular accusative
  
  • for neuter nouns, das is the same in nominative and accusative

So even though the forms look simple, the case is still accusative because of what the verbs are doing.

Because the same subject applies to both verbs.

German often does this when two actions have the same subject:

• Meine Schwester nimmt eine Stecknadel und steckt das Papier fest.

This is more natural than repeating the subject:

• Meine Schwester nimmt eine Stecknadel und meine Schwester steckt das Papier fest.

English does the same thing:

• My sister takes a pin and fastens the paper.

So the subject meine Schwester is understood for both nimmt and steckt.

Because this is a separable verb: feststecken.

In a main clause, German separable verbs split apart:

• the conjugated verb goes in the normal verb position
• the prefix goes to the end

So:

• infinitive: feststecken
• in the sentence: steckt ... fest

That is why you get:

• Sie steckt das Papier fest.

This is the same pattern as:

• aufstehen → Ich stehe um 7 Uhr auf.
• einkaufen → Wir kaufen heute ein.

In this sentence, it is best understood as the separable verb feststecken.

Here feststecken means something like:

• to pin down
• to fasten in place
• to attach firmly with a pin

Plain stecken usually has meanings like:

• to stick
• to put into
• to insert

So das Papier feststecken is not just put the paper somewhere. It means secure the paper in place.

Because the verb is nehmen, and in the 3rd person singular it changes vowel:

• ich nehme
• du nimmst
• er/sie/es nimmt

So meine Schwester = she, and therefore the correct form is nimmt.

This is an irregular verb.
By contrast, stecken is regular:

• ich stecke
• du steckst
• er/sie/es steckt

So the sentence contains:

• nimmt = irregular form of nehmen
• steckt = regular form of stecken

It could be more explicit, but it does not have to be.

The first clause already introduces the pin:

• Meine Schwester nimmt eine Stecknadel ...

So in the second clause, German can leave the instrument understood:

• ... und steckt das Papier fest.

If you want to spell it out, you could say:

• Meine Schwester nimmt eine Stecknadel und steckt das Papier mit einer Stecknadel fest.

or more naturally:

• Meine Schwester nimmt eine Stecknadel und steckt das Papier mit ihr fest.

But since the pin was just mentioned, leaving it out is perfectly natural.

Yes, it is present tense.

German uses the simple present much more broadly than English does. So:

• nimmt can mean takes or is taking
• steckt ... fest can mean fastens or is fastening / pinning down

German usually does not need a special progressive form like English is doing.

So this sentence can describe:

• a habitual action, depending on context
• an action happening right now, depending on context

German main clauses normally follow the verb-second rule.

That means the finite verb comes in the second position:

• Meine Schwester | nimmt | eine Stecknadel
• [Meine Schwester] | steckt | das Papier | fest

In the second part, the subject is not repeated, but the same rule still applies: the finite verb steckt comes early, and the separable prefix fest goes to the end.

So the sentence combines two common German patterns:

1. verb-second word order
2. a separable verb in the second clause

Yes. Separable verbs are normally listed in dictionaries in their full infinitive form, so you would look up:

• feststecken

Even though in a sentence it may appear split:

• steckt ... fest

That is normal for separable verbs. So when learning vocabulary, it is a good idea to store it as:

• feststecken = to fasten/pin something in place

not just stecken + fest separately.