Простите, я ошибся.

Yes. Простите is used in two situations:

• when speaking to more than one person
• when speaking to one person politely

So in Простите, я ошибся, the speaker is being polite or respectful. If you were talking to one friend, a child, or someone you address informally, you would usually say Прости instead.

The ending -те marks the plural or polite imperative.

The basic verb is простить, meaning to forgive.
Its imperative forms are:

• прости — forgive me, sorry, excuse me, said informally to one person
• простите — forgive me, sorry, excuse me, said politely or to more than one person

So Простите is literally something like forgive me, but in everyday Russian it often works just like sorry.

Yes, absolutely. Извините, я ошибся is very natural.

Both простите and извините can mean sorry or excuse me. A rough difference is:

• Извините is extremely common in everyday speech
• Простите can sound a little more like forgive me, sometimes slightly stronger or warmer

In many real situations, though, they overlap and either one is fine.

Ошибся is the past tense of ошибиться.

• ошибиться — to make a mistake, to be mistaken
• ошибся — made a mistake, was mistaken

The -ся is part of the verb. Historically it is a reflexive ending, but here you should learn ошибиться as a whole word. It does not translate neatly as myself in English.

A very useful pair is:

• ошибаться — imperfective
• ошибиться — perfective

Russian often uses the past tense here because the speaker is referring to a completed mistake:

• Я ошибся = I made a mistake
• also often I was mistaken

It sounds natural when you realize you were wrong and want to admit it.

English sometimes uses present wording, like I’m mistaken or my mistake, but Russian commonly uses the past form in this situation.

This is an important difference.

• Я ошибся — I made a mistake / I was mistaken
  One specific, completed error.

• Я ошибаюсь — I am mistaken / I am making a mistake
  More like a present or ongoing state.

So if you have just realized you were wrong and want to apologize, Я ошибся is the usual choice.

Yes. In the past tense, Russian verbs change for gender in the singular.

• Я ошибся — said by a man
• Я ошиблась — said by a woman

Plural is:

• Мы ошиблись — we made a mistake

This is very normal in Russian past-tense verbs.

Sometimes, yes, but я is often kept here.

The reason is that Russian past tense does not show person clearly. Ошибся by itself only tells you that the subject is masculine singular, not whether it is I or he. So:

• я ошибся = I made a mistake
• он ошибся = he made a mistake

Because of that, including я makes the sentence clear and natural.

A simple approximation is:

Прости́те, я оши́бся
pra-STEE-tye, ya a-SHIB-sya

A few notes:

• Простите is stressed on ти
• ошибся is stressed on ши
• the first о in ошибся is unstressed, so it sounds closer to a
• -бся is a consonant cluster that may feel difficult at first; say it slowly, then speed up

If you want to sound more natural, focus especially on the stress:

• прости́те
• оши́бся

The comma separates two parts:

• Простите — the apology formula
• я ошибся — the explanation

It works much like English Sorry, I was mistaken.

So the comma is normal and expected here.