После тренировки я чувствую себя сильнее, чем вчера.

Because после is a preposition that requires the genitive case.

• dictionary form: тренировка
• genitive singular: тренировки

So:

• после тренировки = after the workout / after training

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после фильма = after the film

So the case here is not optional; после normally goes with the genitive.

In Russian, чувствовать себя is the normal way to talk about how you feel physically or emotionally.

So:

• Я чувствую себя хорошо = I feel good
• Я чувствую себя уставшим = I feel tired
• Я чувствую себя сильнее = I feel stronger

If you say just я чувствую, it usually means I feel/sense something as an object:

• Я чувствую холод = I feel cold
• Я чувствую боль = I feel pain
• Я чувствую запах дыма = I smell smoke / I sense the smell of smoke

So in your sentence, себя is needed because the meaning is I feel myself to be...

Себя is the reflexive pronoun, roughly oneself / myself / yourself / himself, depending on who the subject is.

In this sentence:

• я чувствую себя = I feel myself

But in natural English we usually just say I feel.

The useful thing to remember is that себя does not change for person:

• я чувствую себя = I feel
• ты чувствуешь себя = you feel
• он чувствует себя = he feels
• мы чувствуем себя = we feel

So Russian keeps себя, while English usually does not say myself/yourself in this kind of sentence.

Сильнее is the comparative form of сильный (strong).

• сильный = strong
• сильнее = stronger

Russian often forms comparatives with endings like:

• -ее / -ей
• sometimes other changes depending on the adjective

Examples:

• быстрый → быстрее = faster
• умный → умнее = smarter
• сильный → сильнее = stronger

In this sentence, сильнее does not change for gender or number. It stays the same:

• я чувствую себя сильнее
• она чувствует себя сильнее
• они чувствуют себя сильнее

Because the sentence is making a comparison: stronger than yesterday.

If you used сильный, you would just be saying strong, not stronger.

Compare:

• Я чувствую себя сильным. = I feel strong.
• Я чувствую себя сильнее, чем вчера. = I feel stronger than yesterday.

So:

• сильным = strong
• сильнее = stronger

The second one is specifically comparative.

Чем is the usual word for than after a comparative in Russian.

So:

• сильнее, чем вчера = stronger than yesterday
• лучше, чем раньше = better than before
• быстрее, чем я думал = faster than I thought

It introduces what you are comparing with.

In your sentence:

• сильнее = stronger
• чем вчера = than yesterday

Russian often leaves out words that are obvious from the context.

English also does this sometimes:

• I feel stronger than yesterday instead of
• I feel stronger than I felt yesterday

Russian does the same thing:

• сильнее, чем вчера

The full idea is understood automatically, so Russian does not need to repeat чувствую себя or another verb.

Yes, but it means something slightly different.

• Я чувствую себя сильным. = I feel strong.
• Я чувствую себя сильнее, чем вчера. = I feel stronger than yesterday.

The first sentence describes your state without comparing it to anything.

The second sentence compares your current state to an earlier one.

Also notice the form:

• сильным is an adjective in the instrumental case
• сильнее is a comparative

Both are possible after чувствовать себя, but they express different ideas.

Yes, grammatically you can say:

• Я чувствую себя более сильным, чем вчера.

That also means I feel stronger than yesterday.

But сильнее is usually more natural and common in everyday speech.

A simple way to think about it:

• сильнее = the normal, compact comparative form
• более сильным = a more analytical more strong type of structure

In ordinary conversation, сильнее is the better default choice here.

No, Russian word order is fairly flexible.

Your sentence:

• После тренировки я чувствую себя сильнее, чем вчера.

This is very natural and neutral.

But other orders are possible too, depending on what you want to emphasize:

• Я после тренировки чувствую себя сильнее, чем вчера.
• Я чувствую себя сильнее после тренировки, чем вчера.
  (possible, but the focus shifts)

Putting После тренировки first makes the time/context very clear right away: After the workout...

So the original version sounds very natural.

Often yes, if the subject is already clear from context.

Russian verb endings usually show the subject:

• чувствую already means I feel

So in context, you could say:

• После тренировки чувствую себя сильнее, чем вчера.

This sounds natural in conversation.

However, keeping я is also completely normal. It can make the sentence clearer or slightly more emphatic.

Because the speaker is describing their current state now.

• я чувствую = I feel / I am feeling

Even though после тренировки refers to something that happened earlier, the feeling is happening in the present moment.

So the timeline is:

1. the workout happened
2. now the speaker feels stronger

That is why the present tense is used.

It can mean either, depending on context.

• после тренировки can mean after the workout
• or more generally after training / after practice

For example, it could refer to:

• a gym workout
• sports practice
• training session

So the word is flexible, and the exact meaning depends on the situation.

The main changes are in the verb, but себя stays the same.

• После тренировки он чувствует себя сильнее, чем вчера.
  = After training, he feels stronger than yesterday.

• После тренировки мы чувствуем себя сильнее, чем вчера.
  = After training, we feel stronger than yesterday.

Notice:

• я чувствую
• он чувствует
• мы чувствуем

But the reflexive pronoun remains себя in all of them.