I would describe your examples 1, 2, 3 and 5 as cases of fronting rather than inversion. I don't know why you call them confusing. The movement of parts of the clause to the front for purposes of emphasis or focus is well established and has always been possible in English.
In the past, a verb-second rule used to exist; it continues to exist in many Germanic languages like German and Dutch. This means that if the first element is the subject, the next should be the (inflected) verb. Therefore:
- The rain came down. (Subject + Verb + Adverbial)
- Down came the rain. (Adverbial + Verb + Subject)
English has largely abandoned he verb-second rule, and is generally required only if you front the negative element.
- I never said that. (Subject + Negative + Verb + Object)
- Never did I say that. (Negative + inflected Verb + Subject + main Verb + Object)
Your sentences 3 and 5 show this movement. Your sentences 1 and 2 show straightforward fronting.
'Proceed' in sentence 4 is in the imperative mood, where there is no Subject. The second clause represents a subjunctive where indeed inversion is used:
- Should you require assistance, please do not hesitate to get in touch.