The UR can not get partial credit even if the take-off is text book.
Underrotated single jumps get no credit.
Anything else will get some credit; without a fall the net score for an underrotated jump will be positive. I.e., they do get partial credit.
Does a downgraded jump with a good entry, good takeoff, good air position, nondisruptive one-foot landing get score more or less than a non-downgraded jump with a fall?
Depends on the relative base value of the attempted jump and the jump of the same takeoff with one less revolution, and the values of the negative GOEs for that element.
See the scale of values:
http://www.usfigureskating.org/content/200809-SP-SOV.pdf
Basically, for jumps up to double lutz, a fall (with -3 GOE) will result in an element score less than 1.0, so after the 1.0 fall deduction the net effect on the final score of attempting the jump will be a negative value. The same would be true of attempted doubles with both downgrades and falls.
A single downgraded to nothing will earn no points (net -1.0 if there's also a fall; e.g., a bad "waxel")
A double downgraded to a single will earn no more than 0.6 (for an otherwise strong 2A attempt), possibly as low as 0.1, but without a fall deduction the net effect will still be positive
So no matter how badly cheated a double jump might be, as long as the skater remains upright it earns some partial credit whereas a rotated double with a fall ends up with net negative value
For jumps between double axel and triple loop, a fall and -3 GOE will net between 0.0 and 1.0 points of partial credit; for each of these jumps, a downgrade and only -1 GOE (if there are several other strong aspects of the jump and no other negatives) ends up earning more net credit than the rotated jump with the fall
So for any jump up to triple loop, it's worth more to cheat the landing, if you can do it "well," than to fall. For some of these jumps, even downgrades with -2 or -3 GOEs without falls end up netting more points than the rotated jump with fall, but for others once the GOE gets to -2 it's worth less than rotating and falling.
Beginning with triple flip, the base values are high enough that even after -3 GOE and fall deduction there still enough points left that the partial credit for rotating the jump and falling ends up being worth more than the partial credit for standing up smoothly on one foot but getting downgraded.
It's always worth more to underrotate and not fall than to underrotate and fall.