What target numbers would be a certain level of difficulty under this system?

Let me tweak your program a little and graph the results in "At Least" mode:

On this graph, the horizontal axis is the target number and the vertical axis is the probability of rolling at least that number. Thus, we can look at each curve and see where it crosses the point on the vertical axis corresponding to the success rate we want; the corresponding point on the horizontal axis (or, rather, the closest whole number to that point) is the target number we want for that difficulty. If we want more precision, we can also switch back to table mode, which will show the probability of meeting or exceeding each target number as a percentage:

Easy tasks should succeed about 75-80 percent of the time

The black line, corresponding to +1 stat bonus and no (dis)advantage, goes down from 83% at 6 down to 72% at 7. So your target number for easy tasks should be 6 or 7.

Moderate tasks should be successful 50-60% of the time

That's a pretty good match for a target number of 8, which a normal +1 roll will meet or exceed 58% of the time.

Hard tasks should be successful between 25 and 40 percent of the time

That would match a target number of 10, which a normal +1 roll will meet or exceed 28% of the time. (9 isn't too far off from your range either, with a success rate of 42%.)

impossible tasks shouldn’t succeed more than 25% and often more like succeeding below 10-15% or the time even with Advantage and a +3 mod.

A target number of 14 is literally impossible with a +1 mod, and even a +3 mod with advantage only gives a success rate of 20%. If even that's too much, pushing the target number up to 15 drops the success rate even with advantage +3 down to 7.4%.

So, all in all, I'd suggest the same target numbers as HellSaint already did: easy = 6, medium = 8, hard = 10 and impossible = 14.

Of course, in practice you're eventually going to have to playtest this and see if the target numbers feel right in play. But at least those numbers seem like a plausible (and fairly easy to remember) starting point.

This answer assumes the baseline is using a +1 modifier.

In order to find the probability of success, you simply need to sum up the probabilities of a number equal to or higher to your threshold. Advantage or disadvantage shouldn't change your baseline, so you can just do the math for the normal case.

Easy

If you set the threshold at 6, you get a probability of success of 83.33%. At 5 it is 91.66% and at 7 it is 72.22%. There is no number you can use to get anything from 75 to 80, so choose between 6 or 7.

For reference, with advantage, 7 is 90% and 6 is 95%, and with disadvantage it is 47.69% for 7 and 64.36% for 6. Overall, 6 seems a better choice.

Moderate

At 8, you get 56.67% chance of success. At 9 it is 41.67%, and at 7 it is 72.22% as discussed before, so 8 seems to be your best guess.

With disadvantage you get 32% and with advantage you get 80%. Note that it is impossible to fit in your range considering both disadvantage and advantage - 32% is already less than 50% and 80% is already more than 60%.

Hard

Setting the ST at 10 will give you 27.78% of success. At 9, it is 41.67% as already mentioned. Choose between them.

Impossible

With advantage, and +3 modifier, even a 14 ST gives you a 20% probability already. At 15, the probability is 7.5%. So, basically, there is no value that fits your requirement, but 14 is probably your best guess. At 15, even with +3, but without advantage, the probability would be 2.78%, which is very low. At 14, it would be 8.34%.

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The more dice you use, and the smaller the value of them, the harder it is to get a smooth variation in the probability. This can be easily seen from the distribution of 2d6, which peaks at 16.67%. If you actually want to have smooth variations in the probability, you will need to use higher numbered dice or just one dice. As an extreme example, you can get 1% variations using a d100.