Why were elements discovered "out of order" after 1950? I'm referring to the fact that there is one element that corresponds to one proton in an atom, a second that corresponds to two protons, up to, I believe, an element with 115 protons. These elements are listed in the periodic table discovered by Mendeleev in the 19th century, or modern expansions thereof.

By "out of order," I'm referring to a situation where elements were not discovered consecutively by number of atoms, e.g. where an element with 103 protons was discovered before an element with 102 protons, and again with 109 protons before 108, according to this table. I use the qualification "after 1950", that is, the "modern era," because this would have given scientists time to sort out Mendeleev's Table after it gave us a road map to atoms. Perhaps more to the the point, the scientific process is a lot more organized than it was in Mendeleyeev's time.

What were the circumstances of these out-of-order discoveries? Isn't it exponentially harder to discover elements with 103 protons than 102 or 101, etc.? Or wouldn't scientists concentrate on creating the atom with the "next" number of protons, e.g. 116 using advanced technology nowadays? Because this is a process that is different from "randomly" coming across atoms commonly found in nature.
The synthetic trans-uranic elements (the "modern era" elements as you call them) are synthesized by bombarding a certain isotope of one element with a certain isotope of another element with a lot of energy in order to get nuclear fusion. The reason they are "out of order" is that the building blocks of these elements have to be very specific isotopes (extra rich in neutrons for example, somewhat abundant for another). This limits the possible sources considerably and so the heavy elements synthesized (and hence discovered) earlier were those where there was a convenient pair of smaller elements to combine to form the larger ones. You don't just bombard each successive element with protons to make the next one.

A lot of the early discoveries were made by bombarding sources with alpha particles, so there are several discoveries that are increases by 2 protons (e.g. Cm->Cf, Am->Bk). Indeed, the table you linked gives more examples of the pairs of elements used to make the heavy actinides and beyond.