For the first part of the question, the short answer is -
The slightly longer answer to the first part of the question is -
This page on pocket vetoes from the website of the United States Senate makes the point quite clearly:
The Constitution grants the president 10 days to review a measure
passed by the Congress. If the president has not signed the bill after
10 days, it becomes law without his signature.
This is set out in Article 1, Section 7 of the US Constitution (A document written in 1787, ratified in 1788, and in operation since 1789 - just to introduce a little historical context!).
If Donald Trump (or any other President for that matter) never signs a bill that has been passed by Congress AND never vetoes it, it simply becomes law without his signature (unless Congress adjourns during the 10-day period - see below).
The answer to the second part of the question is more complicated. The complications come from how the pocket veto is defined in the Constitution
The problem is that if, in the 10-day period that the President has to sign the bill, Congress adjourns, then the bill dies and the legislation must be reintroduced and passed again when Congress reconvenes. Unfortunately, the Constitution doesn't specify what constitutes an adjournment in this context.
This section of the Constitution is based on the provision contained within constitution of the state of New York (1777). The New York constitution includes the additional proviso:
... unless the legislature shall, by their adjournment, render a
return of the said bill within ten days impracticable; in which case
the bill shall be returned on the first day of the meeting of the
legislature after the expiration of the said ten days.
This didn't make it into the US Constitution. There are references elsewhere in the Constitution to adjournments of differing lengths, nowhere does it specify exactly which adjournments would, or would not, be subject to the pocket veto.
An example of the confusion (and, I think, the first instance of a pocket veto being challenged in the courts) was Senate Bill 3185 which would have allowed Native Americans Washington State to sue for damages resulting from the loss of their tribal lands. The bill was passed to President Calvin Coolidge for signature on 24 June 1926, and Congress adjourned for the summer on 3 July. President Coolidge simply ignored the bill.
The Native American tribal leaders sought to claim their rights in the courts. They asserted that the President's pocket veto was invalid, and that therefore the bill had become law. The United States Supreme Court agreed to hear the case, which became known as the Pocket Veto Case.
In a unanimous decision, The Supreme Court upheld the President's action. The judges found no constitutional distinction among the various types of adjournment.
So, to answer the second part of the question, yes, there have been cases where Presidents have simply ignored bills from Congress. However, since Congress was judged to have adjourned during the President's 10-day period of grace before he has to sign or veto the bill, the bills did not become law.
I could not find an example where a bill from Congress has ever become law without the President's signature.