IOS – On iOS, do “Automatic Downloads” (incl. Music, Apps, Books, Updates) download on public or slow WiFi networks

OSX Server does have several VPN servers in it. You can configure these using Server.app however, if your internal name resolution is broken this is likely to be more trouble.

You'll also need to make sure your Mac Mini's VPN port is available to the internet on a static IP (or dynamic host name).

As an interim step I suggest looking at 'Back To My Mac' which is free as part of iCloud. Technically it uses adhoc ipsec vpn tunnels and gives you an DNS name for each mac set up, effectively bypassing the current internal name resolution issue. e.g. server.12345678.members.btmm.icloud.com Services that you run e.g. web and ssh will accessible.

They're btmm DNS address are not internet routable and don't, generally, require any port forward or static IP addresses. For a couple more technical details check here: https://apple.stackexchange.com/a/53776/46039

Here is a step-by-step guide for you:

1. Temporarily disable any Firewall/Internet Security solution/packet filter on your Mac (like LittleSnitch/Hands Off!/Kaspersky Internet Security etc.)
2. Connect to the administrative interface of your router with a working Mac.
3. Make a note of the internal interface (probably 192.168.0.1 in your case)
4. If your router uses MAC-filters to control access, check that the Mac in question isn't blacklisted or is whitelisted if you use whitelists. To get the MAC-address of a Wi-fi interface, open System Preferences -> Network -> Choose "Wi-fi" -> Advanced -> Hardware -> MAC Address on the Mac without internet access.
5. On the Mac without internet access disable all interfaces in the network preferences pane except the Wi-fi interface by choosing the interface and clicking on the little gear -> "Make Service Inactive". You might have to unlock the lock in the left bottom corner first.
6. Choose the Wi-fi interface and click 'Advanced'
7. Choose the TCP/IP settings, choose to configure IPv4 manually and enter the proper IP-address, netmask and gateway address (enter the IP-address found in step 2 as gateway address). In your example this should be any address not already in use in the range of 192.168.0.2-192.168.0.254 as IP-address, 255.255.255.0 as netmask and 192.168.0.1 as gareway.
8. Choose the DNS tab next to the TCP-IP settings.
9. Enter 8.8.8.8 as DNS server. This DNS server is a public one from Google.
10. Save/apply the settings
11. Open Terminal.app
12. Flush any existing routes by entering route -n flush several times.
13. Flush DNS cache with dscacheutil -flushcache;sudo killall -HUP mDNSResponder. You have to enter your admin password to complete the task. This works with Mac OS X 10.9 (Mavericks) only. If another OS X is installed use the appropriate command.
14. Now ping the gateway address and the DNS server with ping -c 5 192.168.0.1 and ping -c 5 8.8.8.8. The command sends 5 pings to the specified address. To stop pinging - if you didn't include the -c option - just enter ctrlC.
15. If you get a successful answer, quit Terminal. If you didn't get an answer, enter ifconfig en1 and re-check the proper settings of your Wi-fi interface. en1 should be your Wi-fi interface.
16. Return to the network preferences and reenable all other interfaces. Check that the Wi-fi interface is on top of the list. To reorder the list click on the little gear -> "Set Service Order..." and rearrange the interfaces.
17. Now open a browser of your choice and try to connect to an arbitrary site.
18. If you have internet access now, you may replace the Google DNS server with DNS-servers of your ISP or simply add them.

This guide didn't solve the problem.

After some very deep investigations (kudos also to John Keates) we found out that a misconfigured dnscrypt/dnscrypt-proxy was the culprit.