Learn Hardcore Russian: Моя кружка синяя.

моя is a possessive pronoun functioning like an adjective (“my”).
кружка is a feminine noun in the nominative case (“mug”).
синяя is an adjective in its predicative role (“blue”).
Together they form the simple equational sentence “My mug is blue,” with моя кружка as the subject and синяя as the predicate adjective.

Why is there no verb equivalent to есть (“is”) in the English translation?

In Russian, the present-tense copula (the verb есть) is normally omitted in equational sentences. You simply place the subject and then the adjective or noun predicate. The meaning “is” is understood without a separate word.

What case are моя and кружка in, and why?

Both моя and кружка are in the nominative case. The nominative is used for the subject of a sentence. Here the entire phrase моя кружка (“my mug”) is the subject performing no action but being described.

Why is it моя кружка and not мой кружка or моё кружка?

Possessive pronouns in Russian agree with the noun they modify in gender, number, and case.

кружка is feminine singular, nominative.
The corresponding form of is in that slot.So you say , never (masculine) or (neuter).