Halte dich an meiner Hand fest.

Yes. It is an informal singular command addressed to one person.

German imperatives often leave out the subject pronoun, so instead of Du halte dich ... you simply say Halte dich ...

Here, halte is the imperative form of halten. You may also hear the shorter form Halt dich an meiner Hand fest, which is very common in everyday speech.

The full verb is festhalten, specifically the reflexive pattern sich an etwas festhalten.

So the full idea is:

sich an meiner Hand festhalten

In German, many verbs have a separable prefix. Here, fest- is that prefix, and in a normal main clause or command it moves away from the verb.

That is why you get:

Halte dich an meiner Hand fest.

Because festhalten is a separable verb.

• Infinitive: festhalten
• Command: Halte ... fest
• Main clause: Ich halte mich fest
• With another verb at the end: Ich will mich festhalten

So fest moves to the end when the verb is conjugated in a main clause or imperative.

Because this expression is reflexive in German: sich an etwas festhalten.

A literal way to think of it is something like:

hold yourself fast onto my hand

English does not usually say it that way, but German does.
Since the hidden subject is du, the reflexive pronoun is dich.

Compare:

• ich → mich
• du → dich
• wir → uns

Because in this expression, German uses an + dative:

sich an etwas festhalten

So Hand is not a direct object here. It is part of a prepositional phrase, and after an in this usage, the noun is in the dative.

Since Hand is feminine singular, the dative form is:

• nominative: meine Hand
• dative: meiner Hand

So:

• an meine Hand = not correct here
• an meiner Hand = correct

Because mein changes its ending depending on case and gender.

For die Hand:

• nominative: meine Hand
• accusative: meine Hand
• dative: meiner Hand

Since this sentence needs the dative, you get meiner Hand.

So the ending -er is simply the dative feminine form of mein.

Yes, but it means something different.

• Halte dich an meiner Hand fest = Hold on to my hand
• Halte meine Hand fest = Hold my hand tightly

The first sentence is reflexive and means someone uses your hand for support.
The second sentence is transitive and means someone grips your hand firmly.

So the grammar change creates a real meaning change.

No. Festhalten can have several meanings depending on context.

Common ones include:

• to hold tight
• to hold on to
• to keep / stick to
• to record / note down
• to detain / arrest

But with sich an + dative, it specifically means to hold on to.

So in this sentence, the meaning is clearly the hold on sense.

You would change the imperative and the reflexive pronoun:

• informal singular: Halte dich an meiner Hand fest
• informal plural: Haltet euch an meiner Hand fest
• formal singular or plural: Halten Sie sich an meiner Hand fest

So the pattern stays the same; only the command form and reflexive pronoun change.