Just looking at the very first line, ついたち isn’t used for counting days. One day is just ichinichi. Tsuitachi is for the first day of the month, which is a bit different. If you’re aware of that, then we all good, but i wasn’t sure if you were aware of the difference in those two words so figured I’d say something.
That chart basically covers all of the exceptions - aside from that it’s pretty consistently on’yomi+counter. The main exceptions remaining are:
四 and 七 use their kun’ and on’yomi fairly interchangeably, because they like to be weird like that, but there are some cases where one reading or the other is always used. For example, 四人 is always よにん, because しにん sounds like 死人 which means “corpse”. (And on that note, 四 occasionally drops the ん from its kun’yomi, because it’s easier to say.)
Sometimes counters get rendaku’d, and sometimes they get geminated. For example, 匹 (counter for small animals) is read as ひき, but for example when it’s 六匹 the reading is ろっぴき. These follow the standard rendaku and gemination rules, so you’ll get the hang of it over time.
Wikipedia has a pretty good article on the subject:
To be clear, it’s とお with no つ.
The kun’yomi numbers do continue past ten, but you’ll basically never ever see those in modern Japanese. The 個 (こ) counter is used instead, with on’yomi numbers.