For (3), the word you're thinking of is "mane" 'in the morning', which looks very much like an ablative but which doesn't have any other forms. There are definitely other words like that, such as "fas" 'right, propriety, justice'.

For (5), these are called pluralia tantum (singular "plurale tantum").

https://en.wikipedia.org/wiki/Plurale_tantum

I gave some examples of Latin irregularities elsewhere in the thread, and I like your examples too!