Would Vice President Pence take over and become president (in lists of presidents and numbered 46th president) for one week/less than a week?
Impeachment and conviction of the President would remove the President. Section 1 of the 25th Amendment states that: "In case of the removal of the President from office or of his death or resignation, the Vice President shall become President". This suggests that the Vice President would succeed to the office of President.
Would there be an inauguration for him?
Given past precedent regarding presidential succession by the Vice President, all that is required of the new President is to be sworn in by a judicial officer who can administer oaths (like a judge or notary public), so no inauguration is necessary.
Would he get all benefits of a president/ex-president (even if his "term" as president was only some days)?
He has served as a former President and succeeded to the office - relevant laws like the Former Presidents Act (which assigns various benefits) only state that the person in question had to have held the office of President in the past, which would have occurred under Section 1 of the 25th Amendment when the President was removed from office.