Learn Hardcore Portuguese (Portugal): A propina deste semestre é alta, por isso ele guarda cada euro. - The tuition fee this semester is high, so he saves every euro.

In Brazilian Portuguese, propina usually means bribe.
In European Portuguese, bribe is usually suborno (or luvas, informally), and propina in everyday talk about university almost always means tuition fee.

So it’s a classic false friend depending on the variety of Portuguese.

Why is there an A at the beginning: A propina deste semestre? Could I just say Propina deste semestre?

The A is the definite article (the) and it’s natural and preferred here:

A propina deste semestre é alta = The tuition this semester is high.

Leaving out the article (Propina deste semestre é alta) sounds incomplete or like a headline in a newspaper, not normal spoken Portuguese.

In European Portuguese, definite articles are used more frequently than in English, especially with general or specific known things like:

A água está fria. – The water is cold.
O almoço está pronto. – Lunch is ready.

So here you should keep the article: A propina…

What exactly is deste in deste semestre? How is it different from de este or do semestre?

Deste is a contraction of de + este:

de = of / from
este = this (masculine, singular)
→ de + este = deste = of this

So:

deste semestre = of this semester / this semester’s

You don’t say de este semestre; in normal speech and writing it must contract to deste semestre.

Compare:

do semestre = de + o semestre = of the semester (not pointing to “this” one specifically)
deste semestre = of this semester (specific, near in time)

Other examples:

desta semana = de + esta semana – of this week
da semana = de + a semana – of the week

Why is semestre masculine but alta is feminine? What is alta agreeing with?

The adjective alta agrees with propina, not with semestre.

a propina – feminine singular
o semestre – masculine singular

In A propina deste semestre é alta:

Subject: a propina (feminine)
Adjective: alta must be feminine singular to match propina.

If the subject were masculine, the adjective would change:

O semestre é longo. – The semester is long.
O preço é alto. – The price is high.
A propina é alta. – The tuition is high.

So gender agreement is always with the subject noun, not the nearby noun inside a phrase like deste semestre.

Why is it é alta and not está alta? What’s the difference here?

Both é alta and está alta can be possible, but there’s a nuance:

é alta (with ser) suggests a more permanent or defining characteristic:
- The tuition (by nature / generally) is high.
está alta (with estar) would stress a temporary or currently high level, compared to some usual baseline:
- The tuition is (currently) high / higher than usual.

In a context like talking about a university that is known for expensive tuition, é alta is more natural: it sounds like a general fact.

If the idea is “this semester it’s unusually high (maybe more than last year)”, you might hear:

A propina deste semestre está muito alta.

What does por isso mean, and how is it different from então or porque?

Por isso literally means for that (reason) and is best translated here as so or therefore.

A propina deste semestre é alta, por isso ele guarda cada euro.
→ The tuition this semester is high, so he saves every euro.

Differences:

por isso: explicitly gives a logical consequence:
- X is true, therefore Y happens.
então: can also mean so / then, but is more versatile and very common in speech. Sometimes it’s just a filler (so, well).
- You could say: A propina é alta, então ele guarda cada euro. – acceptable in spoken language.
porque means because and introduces the reason, not the result:
- Ele guarda cada euro porque a propina deste semestre é alta.
  – He saves every euro because the tuition this semester is high.

So por isso and porque connect the same two facts but from opposite directions: por isso → result; porque → cause.

Why is it ele guarda cada euro and not ele poupa cada euro? What’s the difference between guardar, poupar, and salvar?

All three can involve the idea of saving, but they’re used differently:

guardar
- Basic idea: to keep / put away / store, and by extension save (money).
- Ele guarda cada euro. – He keeps/saves every euro.
- Very common, slightly more concrete: you “keep” the money instead of spending it.
poupar
- More specifically to save (by not spending / by economizing).
- Ele poupa cada euro. – He saves every euro (is very economical).
- Often used with the idea of being thrifty: poupar dinheiro, poupar energia, poupar tempo.
salvar
- Mainly to save in the sense of rescue (a person, a file on a computer, etc.).
- Not typically used for “saving money” in this context.

In your sentence, guardar is natural and common, but poupar would also make sense, with a bit more emphasis on being frugal.

Why is it cada euro and not cada euros?

In Portuguese, cada is always followed by a singular noun, even though the overall meaning is plural:

cada euro – literally each euro → means every euro
cada pessoa – each person
cada dia – each day / every day

You do not say cada euros or cada dias.

If you want a more obviously plural structure, you’d use todos os:

Ele guarda todos os euros. – He saves all the euros.
Todos os dias – every day / all the days.

Could I omit ele and just say …por isso guarda cada euro?

Yes. In Portuguese, the subject pronoun is often dropped because the verb ending already indicates the person:

(Ele) guarda cada euro. – He saves every euro.

Both are grammatically correct:

por isso ele guarda cada euro – a bit more explicit or contrastive (“he” as opposed to someone else).
por isso guarda cada euro – more neutral, and very natural, especially in written narrative.

In many contexts, omitting ele would sound perfectly normal.

Can I change the word order and put por isso at the beginning of the sentence?

You can, but you’d normally keep the full context:

A propina deste semestre é alta. Por isso, ele guarda cada euro.

Putting Por isso at the very start without the previous sentence would be odd, because por isso refers back to something already mentioned.

Within one sentence, the usual order is cause first, result after:

A propina deste semestre é alta, por isso ele guarda cada euro.

The reverse order (result first, then cause) normally uses porque:

Ele guarda cada euro porque a propina deste semestre é alta. – He saves every euro because the tuition is high.