I read that the new sanctions Congress placed on Russia are virtually VETO-PROOF! How does THAT work?

First, a bill must pass both houses of Congress by a majority vote. After it has passed out of Congress, it is sent along to the President. If the President signs the bill, it becomes law.

The President might not sign the bill, however. If he specifically rejects the bill, called a veto, the bill returns to Congress. There it is voted on again, and if both houses of Congress pass the bill again, but this time by a two-thirds majority, then the bill becomes law without the President's signature. This is called "overriding a veto," and is difficult to do because of the two-thirds majority requirement.

Alternately, the President can sit on the bill, taking no action on it at all. If the President takes no action at all, and ten days passes (not including Sundays), the bill becomes law without the President's signature. However, if the Congress has adjourned before the ten days passes and without a Presidential signature, the bill fails. This is known as a pocket veto.

I read that Trump thinks he can pardon himself.  How is that true? 

You only need to look as far as Article 2, Section 2, Clause 1 of the US Constitution for your answer. 

It says that the President..."shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment." There's no language that prohibits a "self-Pardon". And there's no requirement for Congressional or Judicial review of any Reprieves or Pardons that the President may choose to grant to anyone, including him or herself. The only restrictions there appear to be are that the transgressions be against the United States and they can't be issued for (successful) Impeachments. (Sorry all you ex Federal judges that have been successfully removed from the bench via Impeachment proceedings.) And any attempt by Congress or the Judiciary to "modify" that language would probably be seen as a violation of the Separation of Powers.

Now, on the flip side . . . accepting a Pardon would appear to be an automatic admission of guilt; after all, why accept a Pardon if you've done nothing wrong? (I'm fairly certain that there's case law to back that up.) So if a sitting President were to Pardon him or herself for some specific transgression or "just because" (like Nixon's Pardon, no specific crimes mentioned in it, if I remember correctly) the next logical, and hopefully very rapid step would be for the House to Impeach and then the Senate to conduct the trial unless the President in question resigns before Impeachment proceeding move forward.

I should also point out that the possibility of a President granting a self-pardon is in uncharted waters with the Law of Unintended Consequences being ever present if that does happen. The only thing that can be guaranteed is that it will be most entertaining, for months on end. And it would give Congress yet another excuse to be totally useless. This post was edited by Salt and Red Pepper at July 25, 2017 12:04 AM MDT