It's definitely a good first attempt! I especially liked the first clue. But there are four recurring issues I see:
- Invalid definitions (not matching the answers)
For "acceptable" definitions, I recommend this blog post on how straight crosswords do definitions. Rules 7, 9, 10, and 23 don't really apply to cryptics, but the rest definitely do, and they're important to look over - especially the first few rules.
- Indirect anagrams. These are when you synonymize words and then angram. They're almost always considered unfair, to the point where I gave it as a rule in my guide that they should never be used.
- Violations of what I called the "split rule". Any cryptic clue should be dissectable into a wordplay and definition part, but these two parts should not interact. You should be able to split the clue into its two parts by cutting it once. (Except in the case of &lit clues, which are both rare and insanely difficult to pull off well.)
- Using "head", "top", and similar words to extract more than one letter from another word. Some people might accept this, but I think most wouldn't: typically, "head" refers to the first letter, not any number of letters at the beginning.
Check out the start of this nutritionist's advice! (3)
"This" is an unused word. Also, I would interpret "start" as the first letter: I think you should have used "most of" instead of "the start of" for that part. (That being said, I liked "check out" as a definition! Really clever, and very nicely misleading in the surface.)
I would probably modify this to "Check out most of nutritionists' advice (3)".
Walrus and Carpenter action figures remove their prey's head and scramble it. (4)
This is called an indirect anagram: you're taking a synonymized word and then anagramming it. Indirect anagrams are almost always considered to be unfair, and to make things even worse, you modify this one before doing the anagram! This would make a lot of cryptic solvers pretty mad.
Also, you say "remove the head" but you keep the 'head' instead. (Though again, "head" should only be the first letter.) Your definition is not at the start or end of the clue, so this does not follow the "split rule".
Despondency found in grievous adolescents (3)
The answer is "sad", but your definition clues "sadness". There's a part of speech mismatch between your definition and the answer word, which isn't acceptable in cryptics or in standard crosswords. Definitions and answers must match in part of speech and in tense/conjugation/plurality/whatever. (Ideally, you should be able to find a sentence where you can replace your definition with your answer and keep the same meaning.)
Sucker head is showing amusement (3)
Again, you use "head" for more than one letter (which some may accept and some won't; I try to avoid it), and the definition doesn't match in part of speech (which nobody will accept).
Life of the party's heart activity is abnormal (3)
You use an indirect anagram here. (Also, I'd quibble with "heart activity" here: an EKG measures heart activity, but is not the same thing as heart activity. I wouldn't use "ruler" as a definition for DISTANCE, and I hope you wouldn't either.)
Inside cats are ruling people (4)
"Ruling people" is plural, but TSAR is singular. There's a plurality mismatch here.
Addition imperative without footnotes is coming back (3)
The opposite problem of the other clues: footnotes is plural, implying that more than one letter needs to be removed. (RM)
Animated dog removes actress Sophia's top (3)
First of all, the "head" issue is here again.
But more importantly, the grammar doesn't quite work here: "remove actress Sophia's top" would be acceptable (without that first issue), because it could be interpreted as a command. But "removes" can't: in the wordplay, the one removing things is the solver, not the definition. (And the definition wouldn't be "accessible" by the wordplay anyway, because of the split rule.)
In general, the wordplay for a cryptic clue should always be interpretable as a sequence of commands for modifying words or descriptions for how words have been modified. And the grammar has to match one of those: it may be tempting to change it for a nicer surface, but cryptic reading is king in a cryptic clue. To fix this, you'll either have to change the wording around or find a different method of wordplay entirely.
(Oh, and one other thing I wanted to mention: It's okay for cryptic crossword clues to not sound like perfect English. I've heard jokes that many of them sound more like tabloid headlines. Slightly stilted grammar, or removal of some words as a newspaper would, are totally fine. Don't fall into the trap of sacrificing your cryptic reading for a slightly cleaner surface.)