1/

Alerts do not send immediately informational email (daily report mail of new listings working ok).

conf.: version 8.2.1, sigma, minute cron ok

Is there anything I can do for this, or is it a fundamental bug waiting to be fixed?

2/

There is only a number in the alert listing page. This is very impractical when someone has multiple alerts. Can someone please advise me, where and what piece of code should I insert to display the "search pattern" similar to the picture?

Translated with DeepL.com (free version)

By default all alerts on site are daily and currently no theme provides feature to change this.
Even immediate will most probably has some delay as there must be some cron running to execute it (ideally set minutely cron to run at least once a 5 minutes).

Alerts does not have names, maybe good to consider adding it in future.