Learn Italian: Non ho nessun dubbio.

Because dubbio is a masculine singular noun starting with a regular consonant (d). Just like the indefinite article un, nessuno drops its final -o to become nessun before most consonants and vowels. You only use nessuno before tricky consonant sounds like s+consonant, z, gn, or ps.

Why do we use non if nessun already makes the sentence negative?

Italian requires a double negative when the negative word (like nessun) comes after the verb. Since nessun follows the verb ho, you must put non before the verb. It literally translates to 'I do not have no doubt'.

Could I make this plural and say Non ho nessuni dubbi?

No, nessun and its variations are only used in the singular, even when English uses the plural 'no doubts'. If you want to use the plural noun dubbi, you have to drop the adjective completely and just say Non ho dubbi.

Should there be an apostrophe after nessun here?

No apostrophe is needed here. When nessuno drops its final vowel to match a masculine noun, it is a clean cut (truncation) and takes no apostrophe. The apostrophe is only used for the feminine form before a vowel, like in nessun'amica.