Here are some statements that I had to translate into symbolic form, then negate and express the negation as a positive statement. In each case the assumed domain is given in parentheses.
a. Everyone loves somebody. (All people)
Let:
$L(x,\;y)$ mean $x$ loves $y$.
Symbolic form: $\forall x \exists y [L(x,\;y)]$
Negation: $\exists x \forall y\;[\lnot L(x,\;y)]$
Negated statement in english: Someone doesn't love everyone.
b. Nobody loves everybody. (All people)
Let:
$L(x,\;y)$ mean $x$ loves $y$.
Symbolic form: $\lnot\exists x \forall y [L(x,\;y)]$
Negation: $\exists x \exists y [\lnot L(x,\;y)]$
Negated statement in english:
"Someone doesn't love someone"
or
"At least one person doesn't love someone"
c. If a man comes, all the women will leave. (All people)
Let:
$M(x)$ mean $x$ is a man.
$C(x)$ mean $x$ comes.
$W(x)$ mean $x$ is a woman.
$L(x)$ mean $x$ is a leaves.
Symbolic form: $\exists x {\big[}M(x) \land C(x){\big]} \Rightarrow \forall x{\big[}W(x) \Rightarrow L(x){\big]}$
Negation:
$\exists x {\big[}M(x) \land C(x){\big]} \nRightarrow \forall x{\big[}W(x) \Rightarrow L(x){\big]}$
$\exists x {\big[}M(x) \land C(x){\big]} \land \exists x{\big[}W(x) \land \lnot L(x){\big]}$
Negated statement in english: At least one man will come at least one woman will stay.
d. Not all precious stones are beautiful. (All stones)
Let:
$P(x)$ mean $x$ is a precious stone.
$B(x)$ mean $x$ is beautiful.
Symbolic form: $\exists x\;[P(x) \land \lnot B(x)]$
Negation
$\exists x\;[P(x) \land \lnot B(x)]$ is equivalent to $\exists x\;[P(x) \nRightarrow B(x)]$ where $(a \land \lnot b) \Leftrightarrow (a \nRightarrow b)$
So the negated version is: $\forall x\;[P(x) \Rightarrow B(x)]$
Negated statement in english: All precious stones are beautiful.
e. Nobody loves me. (All people)
Let:
$L(x,\;y)$ mean $x$ loves $y$.
$m$ mean $me$.
Symbolic form: $\lnot \exists x [L(x,\;m)]$
Negation: $\exists x [\lnot L(x,\;m)]$
Negated statement in english: Somebody doesn't love me.
f. At least one American snake is poisonous. (All snakes)
Let:
$A(x)$ mean $x$ is american.
$P(x)$ mean $x$ is poisonous.
Symbolic form: $\exists x [A(x) \land P(x)]$
Negation:
$\forall x\;[\lnot A(x) \lor\lnot P(x)]$
$\forall x\;[A(x) \Rightarrow\lnot P(x)]$; where $(\lnot a \lor b) \Leftrightarrow (a \Rightarrow b)$.
Negated statement in english: No american snake is poisonous.
g. At least one American snake is poisonous. (All animals)
Let:
$S(x)$ mean $x$ is a snake.
$A(x)$ mean $x$ is american.
$P(x)$ mean $x$ is poisonous.
Symbolic form: $\exists x [S(x) \land A(x) \land P(x)]{\big]}$
Negation
$\forall x\;[\lnot S(x) \lor \lnot A(x) \lor \lnot P(x)]$
Grouping together $A(x)$ and $S(x)$:
$\forall x\;{\big[}\lnot [\lnot S(x) \lor \lnot A(x)] \Rightarrow \lnot P(x){\big]}$; where $(\lnot a \lor b) \Leftrightarrow (a \Rightarrow b)$
$\forall x\;{\big[} [ S(x) \land A(x)] \Rightarrow \lnot P(x){\big]}$
Negated statement in english: No american snake is poisonous.
Correction: "Someone doesn't love anybody". (Your symbolic negation is correct though.)
If everybody loves Raymond and Raymond loves only himself, then your suggested statement is True while my correction statement is False.
An orthogonal point worth raising: the given sentence is technically ambiguous, because it could have been alternatively mechanically translated as $\exists x \forall y [L(y,\;x)]$ due to the last word being a hanging quantifier.
The negation (just drop that negation symbol!) ought to be $$\exists x \forall y [L(x,\;y)]$$ "Somebody loves everybody" instead.
No need to change the tense: "A man comes and some woman will stay".
Isn't the symbolic form merely $$\lnot \forall x\;[P(x) \Rightarrow B(x)]?$$
Correct.
The negation (just drop that negation symbol!) ought to be $$\exists x [L(x,\;m)]$$ "Somebody loves me".
f and g are both correct.
Addendum
OP (emphasis mine): The problem I was pointing out is the order of precedence that ¬ has. I used it as to negate just the first quantifier, whereas it seems that you take it to negate the whole expression, even if it's not made explicit by means of parentheses.
Sentences (e.g., ‘the rose is red’, ‘the number of marbles is $7$’, ‘$2x-7=0$’) can be negated. Expressions (e.g., ‘red’, ‘the number of marbles’, ‘$2x-7$’) and quantifiers (e.g., $∀y$) can neither be true nor false, so how can their truth value be flipped, in other words, how can they be negated?
Facts (inserting parentheses to demonstrate that order of precedence is irrelevant): \begin{align}(¬∀x)\:P(x)\;&\equiv\;∃x\:¬P(x)\\&\boldsymbol{\not\equiv}\;∃x\:P(x)\;;\\(¬∀x)∃y\:P(x,y)\;&\equiv\;∃x∀y\:¬P(x,y)\\&\boldsymbol{\not\equiv}\;∃x∃y\:P(x,y).\end{align}
After all, the negation of $$\text{not all xylophones are pink}$$ is $$\text{some xylophone is not pink},$$ rather than $$\text{some xylophone is pink}.$$
If this addendum has revised your understanding, then my above responses to Parts b & d, and my first comment below, will now make more sense.