Сестра жарит омлет с грибами быстрее, чем я.

Yes. Жарит is:

• 3rd person singular
• present tense
• from the imperfective verb жарить

Because it is imperfective, it can describe either:

• an action happening right now: is frying
• a habitual/repeated action: fries

Russian does not have a separate built-in progressive form like English is frying, so context tells you which meaning is intended.

Омлет is in the accusative singular because it is the direct object of жарит.

The reason it looks unchanged is that омлет is:

• masculine
• inanimate

For most masculine inanimate nouns in Russian, the accusative singular is the same as the nominative singular.

So:

• nominative: омлет
• accusative: омлет

That is completely normal.

After с meaning with, Russian normally uses the instrumental case.

So:

• грибы = mushrooms
• грибами = with mushrooms

This phrase means with mushrooms and describes the omelet: an omelet containing mushrooms.

So:

• омлет с грибами = omelet with mushrooms

Here it most naturally describes омлет, not жарит.

So the structure is:

• Сестра = sister
• жарит = fries/is frying
• омлет с грибами = an omelet with mushrooms
• быстрее, чем я = faster than I do

In other words, it means the sister is frying a mushroom omelet, not that she is frying something together with mushrooms as a separate adverbial detail.

Быстрее is the normal comparative form of быстро.

So:

• быстро = quickly
• быстрее = more quickly / faster

Russian usually prefers this simple comparative form in everyday speech.

Более быстро is grammatically possible, but it sounds more formal, heavier, and less natural here.

So быстрее is exactly what a native speaker would normally use.

Because я is understood as the subject of an omitted verb.

The full idea is something like:

• Сестра жарит омлет с грибами быстрее, чем я жарю омлет с грибами.

Since the repeated part is left out, only я remains, and it stays in the nominative case because it is the subject.

So:

• чем я = than I do

Using чем меня here would be wrong.

They are related, but not built the same way.

быстрее, чем я

This uses чем and behaves like a comparison with an omitted clause:

• быстрее, чем я = faster than I do

It strongly suggests an understood verb after я.

быстрее меня

This is the shorter comparative construction, where the compared noun/pronoun goes in the genitive:

• меня = genitive of я

This can also mean faster than me, and in many contexts the meaning is very similar.

In this sentence, чем я is especially natural because the comparison is really between two people doing the same action.

Because чем я introduces a comparative clause-like structure.

Russian normally puts a comma before чем in this kind of comparison:

• быстрее, чем я
• лучше, чем раньше
• больше, чем нужно

So the comma here is standard punctuation.

By contrast, if you used the shorter form быстрее меня, there would be no comma.

Yes. Russian word order is more flexible than English, though the neutral order here is:

• Сестра жарит омлет с грибами быстрее, чем я.

This sounds straightforward and natural.

Other word orders are possible for emphasis, for example:

• Омлет с грибами сестра жарит быстрее, чем я.
  Focuses more on the omelet with mushrooms.

• Быстрее, чем я, сестра жарит омлет с грибами.
  Strongly highlights the comparison.

The basic meaning stays similar, but the emphasis changes.

Russian often leaves out possessives like my, his, or her when they are obvious from context.

So depending on the situation, сестра might mean:

• my sister
• his sister
• her sister
• or simply the sister

If the speaker wants to be explicit, they can say моя сестра.

Also, Russian has no articles like a or the, so bare nouns often look less specific than in English until context makes them clear.

It is specifically to fry or to pan-fry.

So this sentence is more precise than a general English verb like cook.

• жарить = fry
• готовить = cook / prepare food in a general way

So if the Russian sentence uses жарит, the speaker is specifically talking about frying the omelet, not just cooking it in some unspecified way.