No manejes el carro con descuido.

Because this is a negative tú command. In Spanish, negative commands use the present subjunctive form:

• (tú) manejas (present indicative) → negative command: No manejes By contrast, Maneja is the affirmative tú command.

No manejes is directed at tú (informal singular: “you”).
If you wanted the formal singular (usted), it would be:

• No maneje el carro con descuido.
  For ustedes (plural in Latin America):
• No manejen el carro con descuido.

Spanish usually omits subject pronouns when the verb ending already makes the subject clear. Manejes already signals tú, so (tú) is optional:

• No manejes... (most common)
• Tú no manejes... (adds emphasis/contrast, like “You—don’t drive carelessly.”)

Yes. In much of Latin America, manejar is very common for driving a car.
You’ll also hear conducir, which is slightly more formal/neutral in some places:

• No conduzcas el carro con descuido. (also correct for tú)

Spanish often uses the definite article (el/la/los/las) where English might use a possessive (my/your), especially when context makes ownership obvious. So el carro can mean “the car” or “your car/that car,” depending on context.
If you want to be explicit:

• No manejes tu carro con descuido.

Both mean “car,” but:

• carro is very common in many Latin American countries.
• coche is more associated with Spain (though it exists in parts of Latin America too). You might also hear auto in some regions.

Literally it’s “with carelessness,” but it functions like an adverbial phrase meaning carelessly or without due care.
Related options:

• descuidado/a = careless (adjective)
• descuidadamente = carelessly (adverb; less common in everyday speech than con descuido)

Yes, and they’re very natural:

• No manejes sin cuidado. = Don’t drive without care.
• Maneja con cuidado. = Drive carefully. (affirmative command) Between the two, Maneja con cuidado is often the more common “safety reminder” phrasing.

It can sound a bit admonishing because it implies the person might drive carelessly. If you want a softer, more polite tone, you can add:

• por favor: No manejes el carro con descuido, por favor. Or switch to a positive instruction:
• Maneja con cuidado, por favor.

Yes, Spanish allows some flexibility, but the most natural placement is usually what you have:

• No manejes el carro con descuido.
  You can move the phrase for emphasis or rhythm:
• No manejes con descuido el carro. (understandable, slightly marked) Most speakers would keep el carro right after the verb here.