Depois do desporto, lavo as mãos com sabonete e água fria.

Because do is the normal contraction of de + o in Portuguese.

• depois de = after
• o desporto = the sport / sports / exercise
• de + o becomes do

So:

• depois do desporto = after the sport / after sports / after exercise

Portuguese uses these contractions all the time:

• de + a = da
• de + os = dos
• de + as = das

Because desporto is the standard word in European Portuguese.

• desporto = Portugal
• esporte = much more common in Brazil

A learner of Portuguese from Portugal should definitely learn desporto as the normal choice.

It is natural and correct, but there is also a very common alternative:

• Depois de fazer desporto... = After doing sport / after exercising...

The original sentence is fine. But if you want to be more explicit about the activity itself, depois de fazer desporto is also very natural in Portugal.

Because Portuguese often leaves out subject pronouns when they are clear from the verb ending.

• lavo already means I wash
• so eu lavo is not necessary

You would add eu mainly for emphasis or contrast:

• Eu lavo as mãos, mas ele não.
  I wash my hands, but he doesn’t.

Lavo is the 1st person singular present indicative of lavar (to wash).

So:

• eu lavo = I wash

In this sentence, the present tense can describe:

• a habit/routine: After exercise, I wash my hands...
• or a present action, depending on context

Portuguese usually uses the definite article with body parts when the owner is obvious.

So Portuguese says:

• lavo as mãos

where English says:

• I wash my hands

Using as minhas mãos is possible, but it sounds more emphatic, contrastive, or less neutral.

So the normal everyday phrasing is:

• lavo as mãos

No. In Portuguese, lavar as mãos is perfectly natural without a reflexive pronoun.

• Lavo as mãos = I wash my hands

If you say lavo-me, that usually means I wash myself in a more general sense.

So for this sentence, lavo as mãos is the normal structure.

Because Portuguese often leaves out articles after com when talking about things in a general, non-specific way.

So:

• com sabonete = with soap
• e água fria = and cold water

This sounds natural when you mean soap and cold water in general, not a specific soap or a specific water already known in the conversation.

If you wanted to refer to something specific, an article could appear, but here the article-free version is the most natural.

Because fria has to agree with água, and água is a feminine noun.

• água = feminine
• fria = feminine form of frio (cold)

So:

• água fria = cold water

Even though água begins with a, it is still feminine.

Yes. Mão is a feminine noun, even though it ends in -o, which is unusual.

• a mão = the hand
• as mãos = the hands

Its plural is irregular:

• singular: mão
• plural: mãos

This is just something you need to memorize.

Not exactly.

In European Portuguese:

• sabonete usually means toilet soap / hand soap, often the kind used on the body or hands
• sabão is a broader word for soap, often associated with cleaning products or laundry soap

So in a sentence about washing hands, sabonete is a very natural choice.

Yes. Portuguese word order is flexible enough for that.

You can say:

• Depois do desporto, lavo as mãos com sabonete e água fria.
• Lavo as mãos com sabonete e água fria depois do desporto.

Both are correct. Putting depois do desporto first gives a little more emphasis to the time phrase: after exercise.