Meine Mutter backt einen Kuchen mit Mandeln.

Because Mutter is a feminine noun, and the possessive word mein- has to agree with it.

• mein Vater = my father
• meine Mutter = my mother

Here, Mutter is:

• feminine
• singular
• nominative (the subject of the sentence)

So the correct form is meine.

In German, all nouns are capitalized, not just proper names.

So in this sentence:

• Mutter is capitalized because it is a noun
• Kuchen is capitalized because it is a noun
• Mandeln is capitalized because it is a noun

This is one of the most noticeable spelling differences between German and English.

The verb is backen = to bake.
It changes form depending on the subject.

With meine Mutter = she, the verb takes the 3rd person singular form:

• ich backe
• du bäckst / backst
• er/sie/es backt

So Meine Mutter backt means My mother bakes.

In a normal German main clause, the finite verb usually comes in position 2.

So the sentence structure is:

• Meine Mutter = position 1
• backt = position 2
• einen Kuchen mit Mandeln = the rest

This is a basic German word-order rule often called the verb-second rule.

Because Kuchen is the direct object of the sentence, and direct objects often take the accusative case.

Kuchen is a masculine noun, and the masculine indefinite article changes in the accusative:

• nominative: ein Kuchen
• accusative: einen Kuchen

So:

• Ein Kuchen ist lecker. = A cake is delicious.
  (Kuchen is the subject)
• Meine Mutter backt einen Kuchen. = My mother bakes a cake.
  (Kuchen is the object)

You usually learn the gender together with the noun:

• der Kuchen = the cake

Since it is masculine (der), the indefinite article is:

• ein in the nominative
• einen in the accusative

German noun gender is something you generally have to memorize as part of the vocabulary.

Because many masculine nouns in German show the case change only in the article, not in the noun itself.

So:

• der Kuchen
• den Kuchen
• ein Kuchen
• einen Kuchen

The noun Kuchen stays the same.
The article changes, and that is what shows the case.

The preposition mit means with, and it always takes the dative case.

So learners often expect some visible dative marking after mit. But here, Mandeln has no article, so there is no obvious article change to show the dative.

Also, Mandeln is plural:

• singular: die Mandel = almond
• plural: die Mandeln = almonds

So mit Mandeln means with almonds.

Because German often leaves out the article when talking about an ingredient or material in a general sense.

So:

• ein Kuchen mit Mandeln = a cake with almonds
• ein Brot mit Nüssen = a bread with nuts
• Tee mit Zucker = tea with sugar

If you added an article, it would sound more specific:

• mit den Mandeln = with the almonds
  meaning some particular almonds already known in context

It describes Kuchen, not Mutter.

So the meaning is:

• a cake with almonds

It tells you what kind of cake it is.
In other words, mit Mandeln functions like an added description of Kuchen.

No, the normal standard form is backt.

The infinitive is backen, and the usual 3rd person singular form is backt.

You may see variation in some verb forms like du bäckst/backst, but in this sentence the correct standard form is:

• Meine Mutter backt einen Kuchen mit Mandeln.

Yes. That is grammatically possible.

German lets you move elements to the first position for emphasis, but the finite verb still stays in second position:

• Mit Mandeln = position 1
• backt = position 2
• meine Mutter einen Kuchen = the rest

That version emphasizes with almonds more strongly.
The original sentence is the most neutral word order.

There are three important parts here:

• Meine Mutter = nominative
  because it is the subject
• einen Kuchen = accusative
  because it is the direct object
• mit Mandeln = dative after mit

So this one sentence is a good example of how German cases work together:

• subject = nominative
• direct object = accusative
• object of mit = dative