No, Pi-hole starts up on system boot. A following restart wouldn't make much sense.
Maybe it's an issue with the order of your network interfaces coming up.
What does /var/log/pihole.log and /var/log/pihole-FTL.log show?
There is the DELAY_STARTUP option for FTL to, well, do exactly what it says. It may be a solution for you. A debug token would still be very helpful so we can check if your interface is configured in a strange way or something similar.
No, because you would loose your entire DNS cache at this point.
I can't spot any obvious errors in your debug log.
Did you manually restart Pi-hole twice within a minute?
Mar 6 07:32:50 dnsmasq[971]: exiting on receipt of SIGTERM
Mar 6 07:32:50 dnsmasq[22331]: started, version pi-hole-2.84 cachesize 10000
Mar 6 07:32:50 dnsmasq[22331]: DNS service limited to local subnets
Mar 6 07:32:50 dnsmasq[22331]: compile time options: IPv6 GNU-getopt no-DBus no-UBus no-i18n IDN DHCP DHCPv6 Lua TFTP no-conntrack ipset auth cryptohash DNSSEC loop-detect inotify dumpfile
Mar 6 07:32:50 dnsmasq[22331]: DNSSEC validation enabled
Mar 6 07:32:50 dnsmasq[22331]: configured with trust anchor for <root> keytag 20326
Mar 6 07:32:50 dnsmasq[22331]: using only locally-known addresses for domain use-application-dns.net
Mar 6 07:32:50 dnsmasq[22331]: using nameserver ::1#5053
Mar 6 07:32:50 dnsmasq[22331]: using nameserver 127.0.0.1#5053
Mar 6 07:32:50 dnsmasq[22331]: read /etc/hosts - 5 addresses
Mar 6 07:32:50 dnsmasq[22331]: read /etc/pihole/custom.list - 0 addresses
Mar 6 07:32:50 dnsmasq[22331]: read /etc/pihole/local.list - 2 addresses
Mar 6 07:33:11 dnsmasq[22331]: exiting on receipt of SIGTERM
Mar 6 07:33:11 dnsmasq[22461]: started, version pi-hole-2.84 cachesize 10000
Side note:
Your router distributes it's own and another IP as DNS server via DHCP. But it's not Pi-holes IP. Is this on purpose?
I am using pihole on my phone, so I can't run nslookup. However, this time I restarted pihole and using my phone trying to make a few DNS request, then run pihole -d and updated the debug token.
The interesting thing is, under pihole DNS settings, if I set "Listen on all interfaces" then the problem occurs, but if I set "Listen only on interface eth0" then there won't be any problem at all...
Maybe it is related. Note that there isn't a solution yet over there. They describe as well that Pi-hole starts but doesn't reply to queries until it is restarted manually.