Learn English – ‘confusion matrix’ for English phonemes

Sometimes. A few symbols have specific names that are normally used. Many have one or two names that are often used, but you can also describe the sound or the letter.

Some tips:

• IPA symbols that are also English letters can take their English names (e.g. ⟨s⟩ /ɛs/ "ess"). However, you may instead choose to describe the represented phone, when the English pronunciation differs greatly from the IPA one (e.g. ⟨j⟩ perhaps "palatal approximant" rather than "jay").
• ⟨ə⟩ is the schwa, as already pointed out. However, "schwa" is sometimes used more generally to refer to any roughly mid-central vowel, regardless of rounding and precise position.
• ⟨ɛ⟩ and ⟨ɔ⟩ are open E and open O, which can equally describe the letters and their phonetic values.
• Letters that look like Greek letters may be called by the English names for the corresponding Greek letters: ⟨ɛ⟩ epsilon, ⟨θ⟩ theta, ⟨ʊ⟩ upsilon, ⟨β⟩ beta, ⟨ɣ⟩ gamma, ⟨χ⟩ chi, ⟨ɸ⟩ phi.
• Those that represent English phonemes that don't have letters in the English alphabet are named by analogy (nasals and voiceless fricatives /F/ are pronounced /ɛF/, like ⟨m⟩, ⟨n⟩, ⟨s⟩, ⟨f⟩; voiced fricatives are pronounced /Fi/, like ⟨z⟩ (AmE), ⟨v⟩). These are ⟨ʃ⟩ "esh", ⟨ŋ⟩ "eng". Exception: ⟨ð⟩ is apparently called "edh" /ɛð/ (also spelled "eth") and ⟨ʒ⟩ "ezh" even though you would expect *"thee" and *"zhee".
• Some have conventional names based on their shapes: ⟨ɤ⟩ ram's horns.
• Some are always named using a description of their phonetic value: ⟨ʔ⟩ glottal stop.
• Diacritics and attachments often have names (themselves either traditional like "diaeresis" or based on their IPA meanings), which gives names for letters with those attachments. For example: ⟨ç⟩ C with cedilla, ⟨pʰ⟩ aspirated P, ....
• Otherwise, there's probably no special name, so just describe either the letter (e.g. ⟨ɰ⟩ "turned m with leg") or the sound (e.g. "voiceless alveolar trill"), whichever is clearer or more concise.

Aside: /ʃ/ "esh" isn't actually an integral sign (they're different Unicode code points), though people will likely still understand you since it accurately describes the shape.

We don't "drop" the /n/, but it does undergo a change - English nasal consonants assimilate to the place of articulation of the following consonant. Thus rainbow is pretty much always pronounced rai[mb]ow, raincoat is rai[ŋk]oat, etc.

As for how this explains your daughter's spelling, consonant clusters are "difficult" in language acquisition, and young children very commonly simplify them. In addition, syllable codas are less psychologically prominent than syllable onsets (and some very young children just delete them across the board). Here, my guess is that the nasal assimilation process makes it particularly hard to perceive the nasal as a distinct phoneme from the subsequent consonant, so your daughter leaves it out from the spelling.

By the way, there's another major allophonic change with nasals in English that is worth mentioning: they nasalize the preceding vowel. So we pronounce lunch as [lʌ̃ntʃ]. This is hard to perceive, but if you record yourself saying (for example) met [mɛt] and men [mɛ̃n], then clip off the final consonants from the recordings, you'll hear a distinct difference in what is left.

Though I've never read anything about this, I'd speculate that in fast speech - because the vowel is so much longer and more salient than the nasal consonant - most of what alerts us to the presence of the /n/ (perceptually speaking) in a word like lunch is the nasalization of the vowel.