Бабушка сказала, что дрожжи свежие, и первый корж уже готов.

Because the subject is бабушка, which is grammatically feminine. In the past tense, Russian verbs agree with gender and number:

• сказал = he said
• сказала = she said
• сказало = it said
• сказали = they said

So Бабушка сказала means Grandma said, with the feminine past-tense form.

Что here means that and introduces a reported statement:

• Бабушка сказала, что... = Grandma said that...

This is very common in Russian after verbs like:

• сказать = to say
• думать = to think
• знать = to know
• видеть = to see

So the structure is similar to English: X said that...

In Russian, дрожжи is a plural-only noun. That means it is grammatically plural even when it refers to one substance.

So Russian treats it like a plural noun, and everything agreeing with it must also be plural:

• дрожжи свежие = the yeast is fresh

This may feel strange to an English speaker, but it is just how the noun behaves in Russian.

Because свежие agrees with дрожжи, and дрожжи is plural.

Agreement here is:

• singular masculine: свежий
• singular feminine: свежая
• singular neuter: свежее
• plural: свежие

Since дрожжи is plural, the adjective must also be plural: свежие.

In Russian, the verb to be is usually omitted in the present tense.

So:

• дрожжи свежие literally looks like yeast fresh
• but it means the yeast is fresh

The same thing happens later in the sentence:

• корж готов = the cake layer is ready

Russian normally does not use a present-tense equivalent of is/are in this kind of sentence.

Because корж is a masculine noun, singular, in the nominative case, so the ordinal adjective must match it:

• masculine: первый
• feminine: первая
• neuter: первое
• plural: первые

So:

• первый корж = the first cake layer / sponge layer

Готов is the short form of the adjective готовый and is very commonly used in the meaning ready.

In Russian, short-form adjectives are often used predicatively, especially after an omitted is/are:

• корж готов = the cake layer is ready

This is more natural than корж готовый in this sentence.

Agreement for the short form is:

• готов = masculine
• готова = feminine
• готово = neuter
• готовы = plural

Since корж is masculine singular, готов is the correct form.

Good question. Both are adjectives, but they are different kinds of forms here:

• свежие is the full adjective form
• готов is the short adjective form

Russian often uses:

• full forms for ordinary description: дрожжи свежие
• short forms for states, especially words like готов, должен, рад, etc.

So both are normal, but the patterns are different:

• свежий / свежая / свежие usually appears in the full form
• готов is very commonly used in the short form when meaning ready

Уже means already.

It shows that the state has been reached by this point:

• первый корж уже готов = the first cake layer is already ready

It often appears before the word it modifies, just like here. It adds the sense that this has happened sooner than expected, or by now.

Because что дрожжи свежие, и первый корж уже готов is a subordinate clause after сказала.

Russian normally separates this kind of that-clause with a comma:

• Она сказала, что...
• Я знаю, что...
• Мы думаем, что...

So the comma before что is required.

Because both parts belong to the same reported statement:

• что дрожжи свежие
• и первый корж уже готов

In English, this is like saying:

• Grandma said that the yeast was fresh and the first cake layer was already ready.

Russian does not need to repeat что before the second coordinated part. One что can cover both clauses.

Бабушка is in the nominative case because it is the subject of the sentence.

It is the person doing the action of сказала.

Basic form:

• nominative: бабушка
• genitive: бабушки
• dative: бабушке
• accusative: бабушку
• instrumental: бабушкой
• prepositional: о бабушке

Here, nominative is used because Grandma is the subject.

Корж is also in the nominative case.

Even though there is no explicit verb is, it is still the subject of the clause:

• первый корж уже готов = the first cake layer is already ready

So both первый and корж are nominative masculine singular.

Yes. In Russian, that is a complete clause:

• дрожжи = subject
• свежие = predicate adjective
• the present-tense are is omitted

So дрожжи свежие is fully grammatical and complete, just as корж готов is.

Yes. Корж usually means a cake layer, sponge layer, or baked round layer used to build a cake. It does not usually mean a whole cake by itself.

So первый корж уже готов suggests that someone is baking a cake in layers, and the first layer is already done.