IMAP seems to check no matter what, but for POP… Mail>prefsAccount>Advanced, is check auto checked?
Using Connection Doctor or Mail Prefs seems to test them all no matter any setting.
No Connection Doctor or network diagnostics. Nothing like that on the iPad.
Edit: but do have an app on the iPad, System Status, which gives loads of information, much more than I can understand or use. Part of it is similar to Activity Monitor. The same dev has an app, Network Analyzer. But with the latest app store malware scare I hesitate to get it. Besides, no idea if it would give me the information I need for this.