Learn English – Why is the plural of “quiz” spelled with double Z

"Travelling" is not wrong and "Travelling" vs "Traveling" is a "British English" vs "American English" thing as well-explained in the linked Wikipedia article:

The British English doubling is used for all inflections (-ed, -ing, -er, -est) and for the noun suffixes -er and -or. Therefore, British English usage is cancelled, counsellor, cruellest, labelled, modelling, quarrelled, signalling, traveller, and travelling. Americans typically use canceled, counselor, cruelest, labeled, modeling, quarreled, signaling, traveler, and traveling.

I think the linked article, "Why do some words have double consonants while others have only one?" seems to be a good starting point. You have to get yourself familiarized with all those examples.

It is important to note that, in two-syllable words such as happening or entering, etc.

If the stress is on the first syllable, the word gets only one consonant + -ing.

"Travelling" is an exception. There is no rule but has some exceptions.

I'm not a native speaker but I see it this way:

Two consonants in a word give us a different pronunciation like in:

• apple and aple are different in pronunciation [ˈap(ə)l] and [ˈeɪp(ə)l]
• little, better (double t sounds like d)
• ladder, bidder (without the double d, it wouldn't be [a] but [eɪ] instead)
• well (without the double consonants I think it wouldn't be [w], but [v] instead)
• sunny (without double n would be pronounced as [sjuni] instead of [sʌni])
• happy (without the double p would be [hāpi] instead of [hapi])
• watt (without double t would be [wat] instead of [wɒt])

As with vowels i think it will be:

• feet and fit are differently pronounced [fiːt] longer i and [fɪt] shorter i. Same works for teen, beetle, tree.
• hoop, spoon have a longer sounding u. [huːp], [spuːn]. You cant' write spun [spʌn], hup [hʌp], because they are pronounced differently and there is no long u in English as a letter.