Vinga, juga amb nosaltres.

Here vinga is an interjection, not a literal verb meaning come. It is a very common Catalan way to encourage or urge someone, like:

• Come on
• Go on
• All right then

So in this sentence, vinga adds an encouraging tone.

Yes. Juga is the imperative form of jugar for tu.

• jugues = you play / you are playing
• juga! = play!

So the sentence is speaking directly to one person in an informal way.

Catalan usually drops subject pronouns when they are not needed, because the verb ending already shows the person.

• juga already tells you it means you (singular, informal) play!

You could add tu for emphasis, but it is not necessary:

• Vinga, tu, juga amb nosaltres.

That sounds more emphatic, something like Come on, you, play with us.

Because after a preposition such as amb (with), Catalan uses the stressed pronoun form:

• amb mi = with me
• amb tu = with you
• amb nosaltres = with us

Ens is a clitic object pronoun, not the form used after a preposition.

For example:

• Ens veu = He/She sees us
• Amb nosaltres = With us

So amb nosaltres is exactly the form you want here.

Yes, historically it is related to venir (to come). But in sentences like this, learners should think of vinga as a fixed conversational word meaning something like come on or go on, rather than trying to translate it literally as a normal form of venir.

So grammatically it comes from the verb, but in everyday use it often behaves like an interjection.

Because vinga is being used as a separate discourse word at the start of the sentence.

The comma shows a pause:

• Vinga, juga amb nosaltres.

This is similar to English punctuation in:

• Come on, play with us.

Without the comma, it would look less natural in standard writing.

Yes. Juga is the informal singular command used with tu.

If you wanted to be more formal, you would use the vostè form:

• Vinga, jugui amb nosaltres.

If you were speaking to more than one person informally:

• Vinga, jugueu amb nosaltres.

So the original sentence is directed at one person in an informal way.

They come from the same verb, but they do different jobs:

• jugues = present tense statement/question: you play
• juga = command: play!

Compare:

• Tu jugues amb nosaltres. = You play with us.
• Juga amb nosaltres! = Play with us!

That final -a here is the imperative form for tu.

Yes, and it is slightly different in nuance.

• Vinga, juga amb nosaltres. = Come on, play with us.
  This is an encouragement or prompt.

• Vine a jugar amb nosaltres. = Come play with us / Come and play with us.
  This focuses more on coming/joining.

So both are natural, but the original sentence sounds more like urging someone who may be hesitating.

A simple learner-friendly pronunciation guide would be:

• Vinga ≈ VEEN-guh
• juga ≈ ZHOO-guh
• amb nosaltres ≈ ahm noo-zahl-TRES

A few notes:

• The j in Catalan is usually like the s in measure or the g in genre.
• The g in juga is softer than a strong English g in many accents.
• amb often sounds quite short in normal speech.

Pronunciation varies by dialect, but this guide will get you close.