(Post Concern validated for: Win7 x64, Server 2008 R2 and just made because it really had grown to be a PITA by nowadays times)

Maybe it is only me - I am used to deactivate/activate my Network Connection on a "Whence needed"-Base, and in ancient/legacy times I, as an Opera (Presto) User, had no problems with that. Now as I am using Firefox in Conjunction with Opera (Presto) and it is more and more firefox each day (Yeah, I will head to vivaldi if they offer an M2 alternative) it is also more and more that I hate how Firefox deals with network connections:
In Opera 12 I could start the Browser, then activate the Network Connection and once Windows had the Connection established I could just "BROWSE".... With Firefox, whenever I start the Browser before Windows established the Connection, that Browser Sessions doesn't find a working connections for ages... starting Firefox after the Connection is established is no problem...
And for a first start after booting up, that may not be a problem - open Network, open Browser.... but I am also used to disable the network for times I have all Tabs that I need for the next few hours (RTFM, etc.) open in the browser, and then, whence enabling the network connection again it is Opera Presto that has no problems to just continue, but for Firefox the best solution to go on is to just shut it down and start anew... or wait a few "CBM64 Tape load"-Moments before FF recognize the new connection (Open a new Tab and fresh URL may help to fasten the process)...

I Haven't tried any other Browser yet (including IE bundled with OS), is that some Win61+ API/Network behavior that Presto isn't aware of due to its age, or is it just firefox using some weird library functions or whatsoever?

I've actually noticed something similar in Linux. When I'm at home I see no reason to disable my connection, but when I have my laptop with me elsewhere I sometimes do. Firefox can be kind of weird about that sometimes. As you said, I think the "best" solution is to restart Firefox.

@Shandra
Can't confirm, at my end at least.
I usually close the browser when leaving the computer. So I did a test now. I did the test in private mode browsing. (I have plenty of RAM, a fast connection and I hate writings to the HD)
Opened six tabs in Firefox(ESR)/Win7 x64 and deactivated my Internet Connection. Then I reactivated my Internet Connection and continued to browse with Firefox from the previously opened tabs without any problem even so this time with a new IP.
I'm on a Workstation Desktop. I have IPv6 disabled in both, the browser and my OS but IMO this shouldn't have any impact.

@krake Usually there's no problem. It's just some kind of weird fluke that I've experienced two or three times, whereas I turn networking on and off all the time.

Krake : no IPv6 on my side too... at least it's good to hear that it is not a general problem, so there may be a way to solve that issue on my system Just to confirm that the other situation I described is also working flawless on your system -> could you please deactivate the network adapter, then start firefox and after it is up connect to the adapter again and whence connected open an url in FF.

1. opened 6 tabs/sites in Firefox
2. disabled the Network Adapter.
3. enabled the Network Adapter and connected (internet connection got lost because of the deactivated network adapter) to the Internet.
No difference, I can continue to browse instantly from the previously opened tabs/sites without any problem.
********************
1. disabled the Network Adapter
2. launched Firefox
3. enabled the Network Adapter and connected to the Internet.
4. started visiting sites. Works instantly like a charm.
********************

It looks like your browser tries to connect (all the time?) unasked to the internet. It's the only explanation that comes to my mind.
You could eventually try to sniff your network with a decent sniffer like Wirehark.

- Edit -
On a second thought: try to flush the DNS Resolver Cache on next try or see what happens if you stop the DNS Client service.

On a second thought: try to flush the DNS Resolver Cache on next try or see what happens if you stop the DNS Client service.

oops, nearly forgotten my posting here due to a vacation with my family....

1st: Thanks for your Test and
2nd: On all of my Windows Systems I have the DNS service disabled, and where I use fixed IP Addresses also the DHCP Client...

So, I've tried it on my Laptop (also Win7 x64, my Setup) and had the same problem as on my Desktop PC - so if you haven't experienced it, it is probably due to something in my windows setup... and something Opera Presto ain't caring about and Firefox utilizes...
At the moment I really don't have the time to venture further into it, but that is something I should keep in mind and maybe I am going to try some settings whenever it is time for me to write back an old system image (which I ever do before taking an new/updated one)...

Edit: Maybe I should also try the other Browsers I am not using and see where are the commons, where are the differences... (IE, Vivaldi, etc.)... As Presto isn't affected it must be something in my setup and probably ain't related to the router/provider....