1. With D. Vader's frightful chomp, he drew a scream (6,5)
"He drew a scream" coupled with the letter counts made the answer pretty straightforward, and then it was just a matter of reverse engineering: Munch = "chomp," and Edvard = "D. Vader's frightful" (i.e., an anagram of D. Vader)
2. No sex-play for one pursuing sly, twisted layers (10)
On this one, "twisted layers" made me imagine warped sedimentary rock, and then the word "strata" popped into my mind and I immediately jumped to Lysistrata, which led me to repunctuate the first part of the clue as "no-sex play." Then I realized that "twisted" applied not to "layers," but to "sly," indicating it was to be rearranged: "lys"; and "one pursuing" meant "i" came after that.
3. By her very nature, a receiver of dead souls (10)
I think now is a good time to mention that the blurb about the 1DS gave a set of categories that the answers would fall into. Mythology was one of the categories, and I think that helped me determine that I was looking for a name of a female receiver of dead souls here. I think I turned "by" into "per" and realized Persephone fit the bill. Then "by her very nature" became clear as "per se," and then "receiver" revealed itself as doing double duty in the clue by cluing the person and the "phone" part of her name.
4. Will I suffer upset eating screwy food? (7)
I was stuck on this for a little while, wondering if "will I" was cluing "ill" as the first part of the answer, or if there was a seven-letter word for "eating" that I could anagram ("eating screwy," even though it's apparently a law of cryptic crosswords not to require solvers to anagram words not in the clue). Once I considered that "screwy food?" could be the non-wordplay clue (more or less) I started thinking of pasta shapes. Rotini, rotelle... I was led by this to zero in on the "illi" of "Will I," and I realized "Will I suffer" contained "fusilli" backwards. ("Will I suffer" upset, or reversed, eating, or containing, screwy food.)
5. Getting closer and closer to having no sign of illness, mother goes out (10)
I was able to pick this one apart without having an answer: there was probably a 12-letter word that meant "having no sign of illness," which contained "ma," and when those two letters were removed, the remaining word would mean "getting closer and closer." I thought the words could end with -amating and -ating, but then when I looked at the categories in the blurb again after getting and categorizing a few other answers, I realized Math was still unused and immediately thought of asymptotic/asymptomatic.
6. Puppet meant to be headless taken in with captive, an arrogant woman who lost her head (5,10)
Between the letter count and "an arrogant woman who lost her head," this was pretty easy to get. Which is good, because I can't quite work out the wordplay here. "Puppet" = marionette, "meant to be headless" = eant ("meant" headless, i.e., without its first letter), "taken in" indicates insertion. So that gives us "Mari-eant-onette," but I can't figure out how "with captive" means to add an i separate from "eant."
7. Rock is pleasant-sounding (6)
Probably the simplest wordplay in the quiz, at least for me. "Sounding" indicates a homophone. A rock homophonous with a pleasant word? Must be gneiss/nice.
8. Court behaves inappropriately in voting for a harsh sentence (13)
I couldn't think of a long enough word to indicate a harsh sentence (in either sense of the word "sentence") until I looked at "court behaves inappropriately," started anagramming "court," got "truco," then immediately swapped vowels and got elecTROCUtion. And the court anagram is "in voting," which is to say, inserted within "election." (Because of the two T's in the answer, one would have been equally successful with "rocut.")
9. With no failing grade, clever nitwit is an example (8)
Another one where you can get away with not picking up on the cryptic-style wordplay: "clever nitwit" is indeed an example (and, in fact, a literal translation) of "oxymoron." The wordplay here is that a word for clever, minus an F ("with no failing grade") can be attached to a word meaning "nitwit" for the correct answer. Clever = foxy -> oxy; nitwit = moron.
10. For his life's meaning, arrests, odds-on, are needed (7)
This is the one I agonized over. Eventually I was left with only the category Fictional Character, and none of the seven-letter characters I could think of would fit any part of the clue, descriptive or wordplay. I thought "odds-on" probably meant I'd be taking every odd letter in a sequence; eventually I did it to the preceding 14 letters in "meaning, arrests" and got "Maigret." It was pronounceable, but utterly unfamiliar to me. If it was right, Maigret must be some law-enforcement type or a detective or some such. As it turns out, this was all correct, but it was a case of me giving up when I submitted this.
11. Mark, starting crazy row, was performer of crazy song (5,5)
She was the only (5,5) "performer of crazy song" I could think of (Aerosmith and Seal didn't fit the letter count). Then I worked backwards: "mark" and "patsy" can be synonyms, "row" and "line" can be synonyms, and "starting crazy" just means "the first letter of 'crazy.'" I guess.
12. Ill-mannered uncle was naughty, with rear end out in view of the whole world (14)
I got this in trying to think of a 14-letter word meaning "view of the whole world." (Plus, the last cryptic quiz's final answer was "Thorsten A. Integrity," and since the LL homepage contains that German monstrosity, I think it was intended as another tribute to the site.) The word is an anagram ("ill-mannered") of "uncle was naughty" when you drop the y ("with rear end out").