It's called a dieresis. It's used to show that the "a" and the "i" are not to be pronounced as a single sound. So it's pronounced something like "na-eve" and not like "knave" or with the "ai" rhyming with the "i" in "knives".
But in 50 years as a native English speaker/writer, I have never written it like that, and have rarely seen it so either.
Another example is "cooperative" where the second "o" in theory has a dieresis. It's pronounced "coh-op...." and not to rhyme with "loop". Again, I've never written it with the dieresis, and don't recall seeing it like that either.
English uses compound words all the time, but it works differently than Dutch.
There is no consistent rule in where compounds are together or separated.
bedroom, workaround, sunglasses, whiteboard, broadsword, toadstool, Englishman, restroom, bathroom, handkerchief, airplane, redhead, desktop, bypass, overhead, overpass, homeschool, underestimate, underground, ...
They come in many flavours:
Noun adjunct + another noun:
bathroom, sunglasses, rooftop, handkerchief, toadstool, grassroots, Sunday, daytime,
Adjective + noun:
broadsword, redhead, Englishman, blackboard,
Preposition + noun:
outdoors, underground, inside,
Preposition + verb:
workaround, outgoing, ongoing, lockdown, uplift, download, markup,
Meanwhile:
dining room, high school, Chinese man, yellow board, garlic bread, go-to, ...
Also, there are terms that can be either separated or not:
passerby/passer-by, ...
This process is unpredictable. Although if a compound does manage to be made into one word, and becomes the predominant form, the one-word form will often stay the predominant form.
Best Answer
Short answer:
The p does not get doubled in 'hope' because it's followed by the silent/ magic e. It's called magic e because it's silent itself, but it often changes the pronunciation of the preceding syllable. It turns a vowel to a diphthong or a long vowel. The only common exceptions are words that end in ‹ve› (e.g. love, have).
Examples:
See, the e changes the pronunciation.
The final consonant does not often get doubled when there's a silent e at the end of a word.
'Hope' ends in a silent e, so the p doesn't get doubled.
When a word ends with a consonant, it often (not always) gets doubled. In your examples, 'mop' ends with a consonant, so it gets doubled in present and past participles (regular verbs).
Explanation:
Doubling final consonants:
1. CVC (consonant-vowel-consonant) constructions:
Consonants (often) get doubled in CVC combinations (CVC are the final three elements of a word), where the vowels are represented with a single grapheme, not digraphs.
The last C in CVC is the final consonant.
Examples:
W and Y do not often get doubled when they come at the end.
Examples:
2. CVVC and CVCC constructions:
Consonants in CVVC or CVCC combinations (digraphs) do not (often) get doubled.
Examples:
3. Before magic e:
Consonant before magic e does not often get doubled.
Examples:
4. After diphthongs:
Consonant after a diphthong does not get doubled. (Almost all the words that have 'diphthong + consonant' often have magic/silent e after the consonant. In fact, the diphthong is a result of adding 'magic e').
Diphthongs are not long vowels. In a 'long vowel', the shape of your mouth does not change while in a 'diphthong', the shape of your mouth changes because a diphthong is the combination of two different vowel sounds.
Examples:
'Hope' has magic e at the end and has a diphthong before p so the p does not get doubled.
5. In stressed syllables:
In most cases (multisyllabic words, I believe), it depends on stress and does not follow CVC method. When the stress is on the last syllable, the consonant gets doubled.
Examples:
Elicit /ɪˈlɪsɪt/ -> elicited — the T does not get doubled because the last syllable is unstressed.
Interpret /ɪnˈtəːprɪt/ -> interpreted — the T does not get doubled because it's a part of unstressed syllable.
Admit /ədˈmɪt/ -> admitted — the T gets doubled because the last syllable is stressed.
Most words are CVC, though they don't follow the CVC method (e.g. elicit, interpret etc).
(For the letter L, head over to this answer).