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:

• Hat (/hæt/) + e -> hate (/heɪt/)
• Rat (/ræt/) + e -> rate (/reɪt/)
• Mat (/mæt/) + e -> mate (/meɪt/)
• Hop (/hɒp/) + e -> hope (/həʊp/)
• Mop (/mɒp/) + e -> mope (/məʊp/)

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:

• Rob -> r-C o-V b-C -> robbed, robbing.
• mop -> m-C o-V p-C -> mopped, mopping.
• Pin -> p-C i-V n-C -> pinned, pinning

W and Y do not often get doubled when they come at the end.

Examples:

• Bow -> bowed
• Grow -> growing
• Crow -> crowing
• Know -> knowing
• Key -> keyed
• Fly -> flying
• Stay -> stayed etc.

2. CVVC and CVCC constructions:

Consonants in CVVC or CVCC combinations (digraphs) do not (often) get doubled.

Examples:

• Cook -> cooked, cooking — the 'k' does not get doubled.
• Team -> teamed, teaming — the 'm' does not get doubled because it uses a digraph 'ea' to represent the phoneme /iː/
• Deem -> deemed, deeming — the 'm' does not get doubled because it uses a digraph 'ee' to represent the phoneme /iː/.
• Accept -> accepted, accepting — the 't' does not get doubled because the construction is CVCC.
• Mix -> mixed, mixing — the 'x' does not get doubled because it consists of two consonant sounds (/ks/) so it makes the construction CVCC.

3. Before magic e:

Consonant before magic e does not often get doubled.

Examples:

• Love -> loved (the 'v' does not get doubled because it's followed by e)
• Move -> moved
• Time -> timed
• Hope -> hoped
• Race -> raced
• Fade -> faded

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.

• Mop /mɒp/ -> short vowel
• Moop /muːp/ -> long vowel
• Mope /məʊp/ -> diphthong

Examples:

• Hop -> hopped — the P gets doubled because it's preceded by a vowel /ɒ/. On the other hand, hope -> hoped — the P does not get doubled because it is preceded by a diphthong /əʊ/ (the magic e at the end of 'hope' also indicates that).
• Pipe -> piped — the P does not get doubled because it's preceded by a diphthong /aɪ/ (the magic e at the end of 'pipe' also indicates that). On the other hand, pip -> pipped — the P gets doubled because it's preceded by a vowel /ɪ/.
• Rat -> ratted — the T gets doubled because it's preceded by a vowel /æ/. On the other hand, rate -> rated — the T does not get doubled because it's preceded by a diphthong /eɪ/ (the magic e at the end of 'rate' also indicates that).

'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).