Я никому не даю свой пароль.

Никому means to nobody or to no one.

It is the dative case form of никто (nobody). Russian uses the dative here because давать means to give, and the person who receives something goes in the dative:

• давать кому? — to give to whom?

So:

• Я никому не даю свой пароль = I give my password to no one / I don’t give my password to anyone

Yes, it is a double negative by English standards, but in Russian this is normal and required.

With negative pronouns like никто, ничто, никому, никогда, Russian normally also uses не with the verb:

• Я никому не даю...
• Он ничего не знает.
• Мы никогда не спорим.

So in Russian, никому не даю is the correct standard pattern. You would not normally say Я никому даю...

Даю is from давать, the imperfective verb to give. Here it means a habitual, general, or repeated action:

• I don’t give my password to anyone
• I never give out my password

Дам is from дать, the perfective partner. It usually refers to a single completed future action:

• Я никому не дам свой пароль = I won’t give my password to anyone

So:

• не даю = I don’t give / I’m not in the habit of giving
• не дам = I won’t give

Russian often uses свой when the possessor is the same as the subject of the sentence.

Here, the subject is Я (I), and the password belongs to that same person, so свой пароль is the natural choice:

• Я не даю свой пароль = I don’t give my password

Using мой пароль is possible, but свой is usually more idiomatic in this kind of sentence.

A simple way to think about it:

• свой = one’s own
• мой = my

So свой пароль is close to my own password.

Пароль is in the accusative case because it is the direct object of даю:

• даю что? — пароль

Since пароль is a masculine inanimate noun, its accusative form is the same as its nominative form:

• nominative: пароль
• accusative: пароль

So nothing changes in spelling here.

Because даю is the 1st person singular form of давать.

Here are some present-tense forms of давать:

• я даю — I give
• ты даёшь — you give
• он/она даёт — he/she gives
• мы даём — we give
• вы даёте — you give
• они дают — they give

So the ending -ю tells you it means I.

Yes. Russian word order is flexible, and changing it often changes the emphasis, not the basic meaning.

Neutral order:

• Я никому не даю свой пароль.

A version with stronger emphasis on password:

• Я свой пароль никому не даю.

A version without я:

• Никому не даю свой пароль.

All of these can mean roughly the same thing, but the focus shifts slightly.

Yes, very often.

Because даю already shows I, Russian can omit the subject pronoun if it is clear from context:

• Никому не даю свой пароль.

This is very natural in conversation. Including Я can sound a bit more explicit or contrastive:

• Я никому не даю свой пароль.
  = I don’t give my password to anyone.

It is one word: никому.

Negative pronouns like this are normally written together:

• никто — nobody
• ничего — nothing
• никогда — never
• никому — to nobody

So the correct spelling is:

• Я никому не даю свой пароль.

not

• Я ни кому не даю...

A natural stress pattern is:

• Я никомУ не даЮ свой парОль.

Approximate pronunciation:

• ya nee-ka-MOO nye da-YOO svoy pa-ROL'

A few notes:

• никому has stress on the last syllable: -му
• даю has stress on the last syllable: -ю
• пароль has stress on the last syllable: -роль

Also, in careful writing the dictionary form is даёшь, даёт, даём, etc., but in normal typing Russians often write е instead of ё.

Most naturally, it sounds like a general rule or habit:

• I don’t give my password to anyone
• I never share my password

Because давать is imperfective, it often expresses repeated or habitual action.

In context, though, it can also describe a present situation:

• Нет, я никому не даю свой пароль.
  = No, I’m not giving my password to anyone.

So the exact English translation depends on context, but the default sense is usually habitual/general.