The ECMA specification does not specify a bounding complexity, however, you can derive one from the specification's algorithms.

push is O(1), however, in practice it will encounter an O(N) copy costs at engine defined boundaries as the slot array needs to be reallocated. These boundaries are typically logarithmic.

pop is O(1) with a similar caveat to push but the O(N) copy is rarely encountered as it is often folded into garbage collection (e.g. a copying collector could only copy the used part of an array).

shift is at worst O(N) however it can, in specially cases, be implemented as O(1) at the cost of slowing down indexing so your mileage may vary.

slice is O(N) where N is end - start. Not a tremendous amount of optimization opportunity here without significantly slowing down writes to both arrays.

splice is, worst case, O(N). There are array storage techniques that divide N by a constant but they significantly slow down indexing. If an engine uses such techniques you might notice unusually slow operations as it switches between storage techniques triggered by access pattern changes.

One you didn't mention, is sort. It is, in the average case, O(N log N). However, depending on the algorithm chosen by the engine, you could get O(N^2) in some cases. For example, if the engine uses QuickSort (even with an late out to InsertionSort), it has well-known N^2 cases. This could be a source of DoS for your application. If this is a concern either limit the size of the arrays you sort (maybe merging the sub-arrays) or bail-out to HeapSort.

Seems like that would be dependent on the specific engine's implementation.

push() is faster.

js>function foo() {a=[]; start = new Date; for (var i=0;i<100000;i++) a.unshift(1); return((new Date)-start)}
js>foo()
2190
js>function bar() {a=[]; start = new Date; for (var i=0;i<100000;i++) a.push(1); return((new Date)-start)}
js>bar()
10

function foo() {a=[]; start = new Date; for (var i=0;i<100000;i++) a.unshift(1); return((new Date)-start)}
console.log(foo())

function bar() {a=[]; start = new Date; for (var i=0;i<100000;i++) a.push(1); return((new Date)-start)}
console.log(bar());

Update

The above does not take into consideration the order of the arrays. If you want to compare them properly, you must reverse the pushed array. However, push then reverse is still faster by ~10ms for me on chrome with this snippet:

var a=[]; 
var start = new Date; 
for (var i=0;i<100000;i++) {
  a.unshift(1);
}
var end = (new Date)-start;
console.log(`Unshift time: ${end}`);

var a=[];
var start = new Date;
for (var i=0;i<100000;i++) {
  a.push(1);
}

a.reverse();
var end = (new Date)-start;
console.log(`Push and reverse time: ${end}`);