There are two types of "vhosts": real and fake.
Real vhosts are usually called "reverse DNS" (also called rDNS)—it's basically a DNS record for your IP address, pointing to some domain name with a PTR record, which is looked up by the server upon connection (be it ircd, httpd, or anything else). This is the "Looking up your hostname..." notice you see immediately after connecting to IRC.
If a user connects from 2001:db8::42, for example, a DNS lookup is performed for 2.4.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.8.b.d.0.1.0.0.2.ip6.arpa.. If the user connected from 192.0.2.123, the equivalent DNS name will be 123.2.0.192.in-addr.arpa.. The reverse DNS can be changed by whoever controls the IP address block. If you have a single IPv4 address for your home network, you usually have to annoy your ISP to get the rDNS changed.
To avoid spoofing, an ircd (the IRC server program) will compare reverse and forward DNS—that is, it will make sure your rDNS hostname points back to your IP address—before displaying it to other users.
For those who are unable to change their real rDNS, most IRC networks offer "virtual hosts" or "cloaks".
Almost every IRC daemon has the ability to change user's displayed hostname at any time; usually this is restricted to operators, or even to services. For example, with the IRC command /chghost joeuser joes-fake-hostname.net (assuming sufficient privileges).
On most networks this process is automated by services—once an operator assigns you a vhost (cloak), you get it set every time you identify to your services account.
(I'm saying "services", not "NickServ" or "HostServ", because most of the time it is a single program presenting itself as all of the *Servs.)
I believe it happens because the IRC server still thinks for a while that you are online when your internet connection is reset and won't allow you to log in with the same nick. You can explicitly reclaim your stale nick by using the nickserv ghost command - manually or by running a script in your IRC client.
Also, you can try using an IRC bouncer - run it on a remote host that's always online.
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I just noticed that right underneath these options, there is an option saying 'Set as Default'... that worked across all rooms instantly :)