Honestly, this sounds like a sendmail.cf problem, as if it's not configured for non-local email, or it's misconfigured.
The file /var/log/maillog.3 is almost certainly an "old" log file, and therefore, nothing new will get written to it. You didn't mention what Linux distro, BSD or Unix you're using, but the current log file, the one sendmail currently appends to, is probably named /var/log/maillog If /var/log/maillog doesn't exist, you could try stopping and starting sendmail, and seeing if a /var/log/maillog or other file gets created.
Debugging: The old telnet localhost 25 trick is very useful, as you've found, but you can use something like the old BSD Mail program's "-v" option. Try http://heirloom.sourceforge.net/mailx.html for source. Using that, you get to see what the SMTP conversation looks like, which can help.
If you're really desperate, you can run sendmail in a debugging mode:
sendmail -bt -C /whatever/sendmail.cf
You can type in email addresses and see what sendmail does with them.
Best Answer
The two most common causes of crashes are video driver bugs and bad RAM.
You can look for clues in logs in
/var/log. Video problems are logged in/var/log/Xorg.0.log. Problems detected by the kernel are logged in/var/log/kern.logor/var/log/messagesor some other file depending on the distribution, I don't know which file Manjaro uses.However, if your system crashes, it often doesn't get a chance to write to the logs.
Do run a memory test. Install Memtest86+ (Arch Linux has it as a package, so you should have it on Manjaro as well). Reboot and select “memory test” at the Grub prompt and let it run for at least one full pass.
If you suspect a video driver problem, try using the free driver if you were using the proprietary driver, or vice versa.