Learn Russian: Саш, принеси веник, пожалуйста, тут грязь. - Sash, bring the broom, please, there's dirt here.

This is the colloquial truncated vocative case. When addressing someone directly in casual speech, first names and family terms ending in -а or -я often drop the final vowel. It creates a very natural, informal tone.

Can I use this shortened form with formal names or regular words?

No, this is strictly for casual, spoken Russian. It only applies to short-form first names (like Маш for Маша) and familiar family terms (like мам or дядь). You wouldn't use it for full formal names or standard nouns.

Why use the perfective command принеси rather than the imperfective приноси here?

In affirmative commands, the perfective aspect is used to request a single, specific action with a concrete result. You want Sasha to bring the broom once, right now, to clean up this specific mess.

Is there a specific reason to use тут instead of здесь for 'here'?

They are completely interchangeable in meaning, but тут is slightly more colloquial. It perfectly matches the informal, conversational vibe established by the truncated name Саш.