How often do two circuit courts issue contradictory legal opinion (my layman's understanding is that they would tend to use the earlier opinion as precedent, so that shouldn't happen too often?)
If that happens, and Supreme Court doesn't get involved in the issue - for example by refusing to hear either of the two cases - what happens then with the issue as far as subsequent cases arising in legal system?
How often do circuit splits arise?
The number of currently outstanding circuit conflicts on legal questions is probably in the range of many dozen on the low end and several hundred to several thousand on the high end. New circuit splits arise somewhere on some issue many times each year.
It is difficult to quantify this because there are lots of subtle differences of wording or application of the law between circuits that ordinarily are not considered circuit conflicts but could lead to a different result under some highly unusual set of facts. Establishing the existence of a circuit conflict, and becoming familiar with the conflicts that are most promising to see resolved, is one of the particular skills of lawyers who practice before the U.S. Supreme Court.
It is also worth noting that state courts can also enunciate questions of federal law that are presented to them without regard to the rule adopted by the federal circuit in which they are located (federal courts in U.S. territories and possessions are bound by the precedents of their federal circuit on questions of federal law). So, there are 50 states and each of the 13 federal circuits (1-11, DC and Federal) which are directly subordinate to the U.S. Supreme Court and could take a position on a question of federal law that could conflict with another jurisdiction leading to a split of authority that can only be resolved by the U.S. Supreme Court.
What happens when a circuit split is not resolved?
When a circuit split is not resolved courts in each circuit follow the precedents set by their particular circuit. Meanwhile, trial courts and appellate courts in circuits that have not taken a position in the circuit split give particularly careful and reasoned consideration to which side the court should take in an undecided circuit as a matter of first impression.
Unresolved circuit splits are not uncommon and can last for decades on issues that the U.S. Supreme Court deems unimportant.