DNS Loop

I also run dnsmasq on my house edgerouter and also have the server=/.local.mesh/x.x.x.x options in my dnsmasq config on the edgerouter.
I used to see this behavior quite often, but it doesn't really do it anymore and I forget how I fixed it. I tried several things over the years and I don't remember which I settled on. I will have to look at my config again...
One way, I do remember is to create a little script on the edgerouter that will grab the mesh "DNS" from your localnode and then you can have dnsmasq use that file as an additional hosts file and it'll all be happy. The problem with this approach is it is not an automatic process, it'll need updating from time to time, sometimes quite often.

One thing too is the request for the "AAAA" record. A "Quad A" record is for IPv6, not for IPv4, which is what the mesh uses.
What you are seeing there is that your computer is also asking for the IPv6 address of whatever.local.mesh, your edgerouter's dnsmasq is dutifully forwarding the request to your node, like it should, but there is no AAAA record for any .local.mesh address, so your node then asks it's default DNS for the AAAA record, and the loop starts, and yes, while they go back and forth like this almost all other DNS requests grind to a halt. Which then gets the wife yelling "The Intertubes are broke again!"
Same thing is happening when you ask for a non-existant mesh address.

Now that I type all that, I think I remember how I sort of "fixed" it.
I am assuming your "localnode" has your edgerouters ip address set as it's DNS server, right? Stop that.
Why does your localnode need to resolve things on your home LAN, it probably doesn't. Just leave your localnodes DNS set at 8.8.8.8 or whatever it is by default now.
If you need to have your localnode "see out" to the internet, for tunnels or whatnot, you could probably just set it a default route pointing at your edgerouters IP. Then the dnsmasq on the node can query google all day long for AAAA and A records for not-even-there.local.mesh and they'll fail like they should and all will be happy.

I hope that even slightly made sense.
I will double check my config and I can also post the little edgerouter script I have that'll query your localnode for a list of all mesh hosts and then create a "hosts file" out of it.

Here's a little script that'll run on an edgerouter or anything else with python and the needed python modules.
It queries a node for the list of hostnames: http://nodename:8080/cgi-bin/sysinfo.json?hosts=1
It then massages that data into a normal ole "hosts file" that can be used with just about anything...

Well since I can't attach it, it's not an allowed file type and the "<code>" tags aren't working either it seems, here ya go:

#!/usr/bin/env python
from urllib2 import urlopen
import json
import os

#change this IP address to be the address of YOUR localnode
localnodeIP = "10.x.x.x"

#this is where the file goes, use this for the "addn-hosts" directive in dnsmasq
#change name or path as you see fit
#(the "user-data" directory is carried across edgerouter upgrades tho ;) )
meshHostsFile = "/config/user-data/mesh_hosts"

#nothing below here should need changing...
url = 'http://' + localnodeIP + ':8080/cgi-bin/sysinfo.json?hosts=1'
response = urlopen(url)
json_obj = json.load(response)

file = open(meshHostsFile, 'w')
for i in json_obj['hosts']:
       file.write(i['ip']+"\t"+i['name']+".local.mesh\t"+i['name']+"\n")
file.write(localnodeIP + "\tlocalnode.local.mesh\tlocalnode\n")
file.close()
os.system("/etc/init.d/dnsmasq restart")