Een andere methode helpt ons om de woorden eenvoudig te onthouden.

In Dutch, adjectives take an -e ending (weak inflection) when they come after any article before a de-woord (common-gender noun). Methode is a de-woord, so you need andere.

• With a het-woord after een, you would drop the -e (e.g. een mooi huis), but here it’s een andere methode.

With helpen + object + infinitive, Dutch allows you to omit om, but including om makes the purpose clause crystal-clear, especially with a pronoun object like ons.

• Both “helpt ons de woorden te onthouden” and “helpt ons om de woorden te onthouden” are grammatically correct.
• In practice, om is often added for clarity or emphasis.

In subordinate clauses or infinitive constructions with te, separable-prefix verbs remain unseparated. The te is placed between the prefix and the verb stem:

• “onthouden” → “te onthouden”
  In main clauses you would split: “Ik houd de woorden goed vast.”, but not here.

Here eenvoudig is an adverb describing hoe you onthouden (how you remember) – “easily / simply.” Adverbs in Dutch don’t get an extra -e.

• The adjective form (eenvoudige) would modify a noun: “een eenvoudige methode”.

Yes. makkelijk (adverb) is a direct synonym meaning “easily.”

• “Helpt ons makkelijk de woorden te onthouden.”
  eenvoudig is a bit more formal or literally “simple,” while makkelijk is more everyday colloquial.

Using de woorden signals you mean specific words already introduced or understood in context.

• If you speak of words in general, you might drop the article: “Woorden onthoud je soms beter in context.”
  Here, however, you’re referring back to a defined set of words.

helpen is a transitive verb that takes a direct object. Since we (we/us) are the ones being helped, you use the object form ons (us).

• Subject form we would not work here, because we can’t be both subject and direct object of helpt.