iMac 2011 issues

What system are you trying to install? The latest system that model can run is High Sierra 10.13.6. If you're trying to download High Sierra or earlier in the past I've had better luck from the Safe Mode because I evidently have some 3rd party software that interferes with the downloading and installing so try the following:

#1 - boot into Safe Mode according to Start up your Mac in safe mode - Apple Support.

NOTE 1: Safe Mode boot can take up to 3-5 minutes as it's doing the following: 

• • • Verifies your startup disk and attempts to repair directory issues, if needed
    • Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)
    • Prevents Startup Items and Login Items from opening automatically
    • Disables user-installed fonts 
    • Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

#2 - go to the System/Software Update preference pane and begin the download.

#3 - run the installer from the Safe Mode.

The recovery server could not be contacted

This problem is a recent occurrence, "recent" meaning it was first reported within the past several weeks.

Complaints are becoming frequent:

Re: macOS Sierra Recovery

Re: Error Recovery server can't be contacted

Re: iMac A1312 goes straight to macOS Utilities and can’t get it to do anything else

The list is growing by the day. We are waiting patiently for Apple to either fix it or acknowledge that older versions of macOS are no longer available on their Recovery servers.

In the interim, without a Time Machine backup, the only practicable way to that Mac operate again is to boot from its original, grey, System Install DVD or a retail, white, Mac OS Snow Leopard Install DVD. Or, a bootable USB installer that you created on an eligible Mac: Create a bootable installer for macOS - Apple Support.

Ah, the vintage 2011 iMac! You're keeping it old school, I see. Kudos on hanging onto a decade-old piece of tech and expecting it to run like it's fresh off the assembly line. But hey, these machines are workhorses, right? Until they're not.

Let's talk about your factory reset woes. First of all, the messages you're getting make it sound like your iMac is trying to reach out to Apple servers to get additional stuff it needs to complete the installation. And, surprise, surprise, those servers aren't playing nice with your seasoned veteran of a machine.

Option 1: Time Machine

If you're fortunate enough to have a Time Machine backup, use it. It's like a lifeboat for your sinking digital ship. Restore and sail away.

Option 2: Bootable USB

You could also create a bootable USB drive with the macOS version that best suits your iMac, then install from that. Ah, yes, the wonders of bootable media.

Option 3: macOS Recovery Over the Internet

Alternatively, try Internet Recovery. Reboot while holding down Command-Option-R and cross your fingers. Maybe this time the recovery server will decide to acknowledge your antique's existence.

Option 4: DNS Trick

Some users have had luck changing their DNS settings to Google's public DNS (8.8.8.8 and 8.8.4.4) to give that server a nudge. Who knows? Your iMac might be more compatible with Google than with its own kind.

Option 5: Update the date and time

Yeah, you read that right. Sometimes, this server error could be because your system's date and time are off. Open Terminal in recovery mode and type "date" to check. If it's out of whack, set it straight. Maybe then the server will stop giving your iMac the cold shoulder.

If all else fails, you could always turn it into an oversized paperweight. Or a museum piece. Your call.