Is it possible/common to say:
I'd like an apple two times.
No. "Two times" in this case does not make sense, because you specify "an apple."
When you use the verb "to like" in the context of eating food, you're implying a further verb, "to eat." "I'd like an apple," in English, implies "I'd like to eat an apple."
At the same time, "an apple" implies a single apple.
If you add "twice" or "two times" to that sentence you end up with:
I'd like to eat an apple twice.
That implies that you're eating the same apple twice, which is not (presumably) what you want to say. Instead, you say:
I'd like to eat two apples.
You can achieve the same effect by changing "an apple" to "apples":
I'd like to eat apples twice.
That suggests eating one or more apples on two occasions. Eating one or more (non-specific) apples is something that can be done multiple times, so the problem doesn't arise.
No, that usage is certainly not common.
You could say
I've had apple two times today
that usage tends to be 'in the past'
For a future usage you could have
I would like to have apple two times tonight
Twice would be a better way to say it
I've had apple twice today
but if you get above three (Thrice) then you start running into awkward words that aren't really in the 'once, twice, thrice' series and reverting to n times is all you can do.
A similar speech pattern is common for exaggeration or emphasis. If you really want to eat a specific apple (it looks, smells or tastes appetizing; you're really hungry; or it's just your favorite kind of apple), you could say "I would like to eat that apple twice". The brevity of 'twice', along with the stressed pronunciation, compared to 'two times' emphasizes that you are aware of the paradox and really, really want to eat the apple. The paradox is that eating implies a complete event which cannot be repeated.
This is a variation of the old pattern "... and twice on Sunday." The exact origins of this phrase are unclear, but it can emphasize your desire for the object. It was incorporated into a love song by the Andrews Sisters in the 1940s has been worked into common language for quite some time.