Je ne veux pas de mayonnaise avec mes frites, et Marie non plus.

In French, after a negated verb, de / d' usually replaces the indefinite or partitive article.

So:

• Positive: Je veux de la mayonnaise. = I want some mayonnaise.
• Negative: Je ne veux pas de mayonnaise. = I don’t want any mayonnaise.

This is a very common pattern:

• J’ai du pain → Je n’ai pas de pain
• Il boit du lait → Il ne boit pas de lait

So de mayonnaise is exactly what you expect after ne ... pas.

Ne ... pas is the standard way to make a sentence negative in French.

Here, the verb is veux (I want), so the negation goes around it:

• Je veux = I want
• Je ne veux pas = I do not want

This is the normal pattern with most verbs:

• Il mange → Il ne mange pas
• Nous savons → Nous ne savons pas

In everyday spoken French, people often drop ne and say:

• Je veux pas de mayonnaise

But in standard written French, ne ... pas is the correct full form.

Because the subject is je (I), and the verb vouloir has to match that subject.

Present tense of vouloir:

• je veux = I want
• tu veux = you want
• il/elle/on veut = he/she/one wants
• nous voulons = we want
• vous voulez = you want
• ils/elles veulent = they want

So:

• Je veux = correct
• Je veut = incorrect

French often uses a possessive where English would not.

Here, mes frites means my fries, but in natural English we would usually just say fries or my fries depending on context. In French, using mes sounds very normal because it refers to the fries that belong to the speaker’s meal or portion.

This happens a lot in French:

• J’ai mal à la tête = I have a headache
  (literally not my head, but French structures possession differently here)
• Il a pris son manteau = He took his coat

So avec mes frites is a natural way to say with my fries.

Avec simply means with, and here it expresses accompaniment: mayonnaise served together with the fries.

So:

• de la mayonnaise avec mes frites = mayonnaise with my fries

That is the most natural choice in this context.

You might sometimes hear sur les frites if you specifically mean on top of the fries, but that changes the focus.
Avec is broader and more natural for ordering or talking about food combinations.

Non plus means neither or not ... either in a negative context.

So:

• Je ne veux pas de mayonnaise ... et Marie non plus.
• = I don’t want mayonnaise, and Marie doesn’t either / neither does Marie.

It is the negative counterpart of aussi:

• Positive: Je veux de la mayonnaise, et Marie aussi. = I want mayonnaise, and Marie does too.
• Negative: Je ne veux pas de mayonnaise, et Marie non plus. = I don’t want mayonnaise, and Marie doesn’t either.

A very common expression is:

• Moi non plus = Me neither

Because French, like English, often leaves out words that are already understood.

Full version:

• Je ne veux pas de mayonnaise avec mes frites, et Marie ne veut pas de mayonnaise avec ses frites non plus.

That is grammatical, but repetitive.

So French shortens it to:

• ... et Marie non plus.

This kind of omission is very common and natural.

Not in this sentence.

Because the first part is negative (Je ne veux pas...), French uses non plus, not aussi.

Compare:

• Je veux de la mayonnaise, et Marie aussi. = I want mayonnaise, and Marie does too.
• Je ne veux pas de mayonnaise, et Marie non plus. = I don’t want mayonnaise, and Marie doesn’t either.

So aussi goes with positive ideas, and non plus goes with negative ones.

Because frites usually means French fries, which are normally thought of as a plural food item.

• une frite = one fry
• des frites = fries

So in this sentence, mes frites means my fries.

The singular frite exists, but you usually use it only if you are talking about one individual fry.

Yes, absolutely. It sounds natural.

In everyday speech, many speakers would make it a little more informal by dropping ne:

• Je veux pas de mayonnaise avec mes frites, et Marie non plus.

But the full written version:

• Je ne veux pas de mayonnaise avec mes frites, et Marie non plus.

is standard, correct, and completely natural.

Yes. Je ne veux pas is direct: I don’t want.
Je ne voudrais pas is softer and more polite: I wouldn’t like or I would not want.

So:

• Je ne veux pas de mayonnaise = straightforward, normal
• Je ne voudrais pas de mayonnaise = more polite, more gentle, often used when ordering

Both are correct. The sentence you have is simple and direct.