Learn Russian: Мам, скажи Маше, чтобы она погуляла с собакой. - Mom, tell Masha to walk the dog for a bit.

Мам is a colloquial, shortened form of мама used specifically for addressing someone directly. This is called the "truncated vocative" case. You create it by dropping the final vowel of a name or family term ending in -а or -я. It makes the request sound much more natural and affectionate in spoken Russian.

Could I also use a shortened form for Masha here, like скажи Маш?

No, because Masha isn't the person being spoken to. The truncated vocative is strictly for addressing someone. Here, Mom is the person being addressed, while Masha is the person receiving the message. So, Маше takes the regular dative case because it answers the question "tell to whom?".

Why is the verb погуляла in the past tense if the English translation is "to walk"?

In Russian, when you report a command or express a desire for someone else to do something, you use the conjunction чтобы (so that / in order to) followed by a verb in the past tense. Even though the action of walking the dog hasn't happened yet, the strict grammatical rule for чтобы in this type of clause requires the past tense.

Why use the perfective verb погуляла instead of the imperfective гуляла?