После тренировки я едва дохожу до дома.

Because после requires the genitive case.

• Base form: тренировка
• Genitive singular: тренировки

So после тренировки means after the workout / after training.

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после тренировки = after training

Едва means barely or hardly.

In this sentence, it shows that getting home is difficult because the speaker is very tired after training.

So я едва дохожу до дома means something like:

• I can barely make it home
• I hardly manage to get home

A close synonym is еле.
Еле is a bit more conversational, while едва can sound a little more neutral or literary.

Because доходить means to reach, to make it to, or to get as far as a place.

Compare:

• Я иду домой = I am going home
• Я дохожу до дома = I make it home / I reach home

With едва, the second verb works especially well, because the idea is not just going, but barely managing to reach the destination.

The prefix до- often adds the meaning of reaching an endpoint.

Дохожу is the 1st person singular present tense form of доходить.

So:

• infinitive: доходить
• I: дохожу
• you: доходишь
• he/she: доходит

Its perfective partner is дойти.

That gives you this contrast:

• доходить = imperfective, repeated/process meaning
• дойти = perfective, one completed act of reaching

So:

• После тренировки я едва дохожу до дома = After training, I can barely make it home / I barely make it home
• После тренировки я едва дошёл / дошла до дома = After training, I barely made it home

With доходить / дойти, the destination is normally expressed with до + genitive.

So:

• дойти до дома
• дойти до остановки
• дойти до конца

By contrast, домой is very common with simpler motion verbs like:

• идти домой
• ехать домой
• бежать домой

So до дома is used here because the verb доходить naturally goes with до.

It looks the same, but here it is functioning differently.

In до дома, дома is the genitive singular of дом, required by the preposition до.

So in this sentence:

• до дома = as far as home / to the house / home

That is different from the adverb дома meaning at home:

• Я дома = I am at home

Same spelling, different grammar.

Yes, most likely.

The present tense here usually describes a regular or typical situation:

• After training, I can barely make it home
• After a workout, I barely get home

It sounds like this happens repeatedly.

If you wanted to talk about one specific past time, Russian would usually use the past:

• После тренировки я едва дошёл до дома
• После тренировки я едва дошла до дома

If you wanted a future meaning, you would use the perfective future:

• После тренировки я едва дойду до дома

Not always, but it is normal and natural.

Russian often drops subject pronouns when the meaning is already clear from the verb form. So you could also say:

• После тренировки едва дохожу до дома

That still clearly means I barely make it home after training, because дохожу already shows 1st person singular.

Including я makes the sentence a bit more explicit and neutral.

Yes. Russian word order is flexible, and changing it usually changes focus more than basic meaning.

The original sentence:

• После тренировки я едва дохожу до дома

This sounds natural because it starts with the time context: after training.

Other possible orders:

• Я едва дохожу до дома после тренировки
  Neutral, but it puts I first.

• До дома я едва дохожу после тренировки
  More emphatic or marked; it highlights home.

In the original sentence, the structure is very natural for everyday Russian:

1. time/context
2. subject
3. adverb
4. verb
5. destination