Recommended watch tool
 

Platform: Asus RT-AC88U running Asuswrt-Rmerlin 380.65

I'm looking for a recommendation on a tool that can ...

1. Scan log files and send an e-mail based on criteria. For example, errors / warnings, port scans, etc.
2. Check the network connections on the router and send an e-mail when one of them isn't working.

I thought about using a cron job running grep and sendmail, but if there was one tool that could do all of the above, that would be preferable.

Welcome.

That one tool would be called a shell script. You can have it use grep, awk, ping, traceroute, dig, and other utilities. It's easy enough and you'll get exactly the functions you need and the messages formatted exactly the way you want.

snmp?

Logwatch
You don't define what you mean by "check the network connections on the router", so it's hard to say. If you mean a basic ping, a shell script is the way to go. If you mean check something more detailed, then SNMP (as Habitual suggested) would be good; scriptable along with snmpwalk too. Otherwise, if you want to check specific routes/traffic/ports/CPU and memory, etc., then who knows?

logwatch! Of course.

I am not worthy. :)

How would you install Logwatch on a platform like Asuswrt ?

So far I've only installed Entware-NG packages.