To what extent (if at all) can we affect our own emotions? My answer depends on who we are. If “we” is a name for all our many inward layers and levels—”My name is legion,” said the demoniac (Mark 5:9)—then obviously our current emotional state will be “affected” (effected, in fact!) whenever we experience something in our environment that we like or dislike, and whenever a given memory (whether happy, sad, or something else) rises up to the surface of consciousness.
I suspect, however, that what you have in mind instead is the possibility of gaining a certain amount of control over emotional states by deliberately positioning ourselves at a point above both outward events and circumstances and the inward flux of reactions. So your question becomes: to what extent can we deliberately choose how we “feel” (or in fact whether we feel) rather than being mechanically driven or prompted to feel a certain way?
Well, if you’re asking what is humanly possible, my answer would be: to a very great extent indeed. To be a man is by definition to be able to pass altogether beyond the usual “flux” of life, both outward and inward, and to share in God’s freedom from what you called the “construct of time”. This, of course, is what the Christian tradition calls theosis or deification; what Hindus call moksha or liberation; what Sufis call fana or extinction.
On the other hand, if you’re asking about what “we” are capable of at the moment—you and I as we are right now—then my response would have to be considerably more cautious and restrained, though I hope not discouraging. Voluntarily placing ourselves outside the flux, taking up a position above our accustomed reactions, is obviously much easier said than “done”! An intense patience is called for.
And do note the quotation marks in that last line: unless the traditions are much mistaken, quod absit, it’s not so much a “doing” that’s needed as an undoing—a refusal to act, or react, in our habitual, mechanical ways; and then, having failed as we almost certainly will not-to-do, repeated refusals to react to our failures. This work is extremely difficult, say the masters, but it’s not impossible, “for it is God which worketh in you” (Philippians 2:13). One unmoving step at a time.