Dum tiu periodo ni marŝis multe.

Here dum means during.

So Dum tiu periodo = During that period.

A useful extra point: dum can also mean while when it introduces a whole clause.

• Dum tiu periodo ni marŝis multe. = During that period, we walked a lot.
• Dum ni marŝis, pluvis. = While we were walking, it was raining.

So dum can work with either:

• a noun phrase: dum tiu periodo
• a clause: dum ni marŝis

Because dum is a preposition, and after ordinary prepositions Esperanto normally uses the basic form without -n.

So:

• dum tiu periodo = during that period

Not:

• dum tiun periodon

The -n ending is not used here because dum already shows the relationship.

A related point: Esperanto can sometimes express duration without a preposition, and then you may see -n:

• Ni marŝis tiun periodon. = We walked for that period.
  This is possible in Esperanto, but dum tiu periodo is clearer and more usual for learners.

Because tiu already means that, so it already makes the noun definite.

• tiu periodo = that period

Esperanto does not normally combine la with demonstratives like tiu.

So you say:

• tiu periodo
• ĉi tiu periodo

Not:

• la tiu periodo

Tiu means that.

It is a demonstrative word used to point out something specific.

• tiu periodo = that period
• tiu libro = that book
• tiu homo = that person

If you want to emphasize this, Esperanto often uses:

• ĉi tiu periodo = this period

So in your sentence, tiu periodo refers to a specific period already understood from context.

Marŝis is made of:

• marŝ- = the root
• -is = past tense verb ending

So:

• marŝi = to walk / to march
• marŝas = walk / walks / am walking / are walking
• marŝis = walked
• marŝos = will walk

In Esperanto, every finite verb has a fixed ending:

• -as present
• -is past
• -os future
• -us conditional
• -u command / wish

So ni marŝis simply means we walked.

It can suggest either, depending on context, which is why English speakers sometimes pause at this word.

In many ordinary contexts, marŝi is used for walking on foot:

• Ni marŝis multe. = We walked a lot.

But in some contexts it can also feel like march:

• La soldatoj marŝis. = The soldiers marched.

So in your sentence, unless the context is military or very formal, marŝis multe is naturally understood as walked a lot.

Because multe is an adverb, and here it modifies the verb marŝis.

• ni marŝis multe = we walked a lot

It tells you how much the action happened.

Compare:

• multa = much / a lot of, describing a singular noun
  
  • multa laboro = much work
• multaj = many, describing a plural noun
  
  • multaj tagoj = many days
• multe = a lot / much, modifying a verb or functioning adverbially
  
  • li laboras multe = he works a lot

So multe is correct because it describes the action of walking, not a noun.

Yes. Esperanto word order is fairly flexible.

The original sentence:

• Dum tiu periodo ni marŝis multe.

You could also say:

• Ni marŝis multe dum tiu periodo.

Both mean essentially the same thing. The difference is mainly one of emphasis or style:

• Dum tiu periodo... puts the time frame first
• Ni marŝis multe... puts the action first

Because Esperanto marks grammatical roles clearly, changing word order is often possible without changing the basic meaning.

A helpful pronunciation guide:

• marŝis ≈ MAR-shis
• the ŝ sounds like English sh

Letter by letter:

• m = m
• a = like a in father
• r = a rolled or tapped r if possible
• ŝ = sh
• i = like ee in see
• s = always plain s, never z

So marŝis sounds roughly like MAR-shees, but with a clear final s, not z.

Tiu is usually pronounced roughly like TEE-oo, said smoothly, almost as one syllable sequence.

More exactly:

• ti = tee
• u = oo

So:

• tiu ≈ TEE-oo

In normal speech, the two vowels flow together.

Yes. That would also be correct.

• dum tiu tempo = during that time
• dum tiu periodo = during that period

The difference is mostly nuance:

• tempo is more general: time
• periodo suggests a more defined stretch or phase: period

So periodo can sound a bit more specific or formal, but both are natural depending on context.

Because none of the words need to be plural.

• periodo is singular because it refers to one period
• ni already means we, so it is inherently plural
• marŝis does not change for singular or plural subjects in Esperanto
• multe is an adverb, so it does not take plural endings

This is an important Esperanto feature: verbs do not change depending on the subject.

So:

• mi marŝis = I walked
• li marŝis = he walked
• ni marŝis = we walked
• ili marŝis = they walked

The verb stays the same.

Yes, it is very natural.

Dum tiu periodo ni marŝis multe is a straightforward, idiomatic Esperanto sentence:

• dum for time frame
• tiu periodo for a specific period
• ni marŝis for past action
• multe for a lot

A very similar natural variant would be:

• Ni multe marŝis dum tiu periodo.

Both are good Esperanto.