Which is the best router

Do you have line of sight? That means can you see from one end to the other. If there is an obstruction in the way, point to point wireless isn't going to work.
Is this system just for connecting one notebook computer or do you plan to add more remote clients?
Which end has the 24dBi antenna? What type of antenna? How much coax cable and what type? If it's a dish antenna, do you know that the beamwidth of the antenna is only about 7 degrees and aiming even at 60 meters is tricky?
I read that to mean that it barely works even when you're close to the DI-624 and with no range extender. Something is wrong here. Before you blunder onward, I suggest you make sure that the HP NX8220 laptop and the DI-624 wireless router can connect reliably and with good speed. That means take a walk while connected to the internet. Leave the big dish and range extender out of the picture. You should be able to go about 20 meters away, with no obstructions, and maintain a solid, reliable, and totally stable connection to the internet. If not, try to figure out if it's the DI-624 or the NX8220 end by comparing it with another known working laptop or connecting to a public hot spot. If the connection is flakey, difficult to maintain, and erratic, it may also be interference from other networks or non-802.11 sources such as cordless phones. The dish antenna will help with these later, but it should be identified.
I'm not going to try and troubleshoot range extender compatibility issues. Basically, you have to have the same chipset in each device for this to work. My luck with mixed chipset range extenders and repeaters has been dismal.
Are you running the latest greatest firmware: | No, the notebook says its connection to the range extender is good. Can you connect to the build in web server in the DI-624 via the DWL-G710 range extender? If not, then the range extender is the problem.
Baloney. If it's all in the same room, you should have superior signal stength. Sounds more like a great excuse to not have you return the DWL-G710 and ask for a refund.
Huh? Putting a highly directional antenna on the range extender is a waste of time. You should get spectacular range in exactly one direction from the range extender, and nothing in any other direction.
I don't think you're having an RF problem. I think there's some manner of fundamental incompatibilty between the various boxes. | your DI-624 rev C3 hardware with firmware 2.50 or higher?
I don't know of any wireless link that will go through part of a house. However, at 60 meters, with a 24dBi antenna at the router, and a 12dBi antenna at the client, there should be enough signal to "drill" through the obstructing house if there are windows in the line of sight. I've reliably gone through an office building with high gain antennas. Remove the range extender and concentrate on the link path.
Interference. That's why store and forward repeaters and range extenders are in my opinion a bad idea.
The "confused" connection is probably due to you having Windoze Wireless Zero Config set to "connect to best connection" or something like that. Uncheck the box and manually select the desired connection.

If the office and home share the same power wiring, another option that might work better is the Netgear 54 Mbps Wall-Plugged Wireless Range Extender Kit Model WGXB102 .
You can get better range with MIMO technology (e.g., D-Link DI-624M and DWL-G650M ), although I don't know if it would be enough to solve your problem.

Hi Jeff,
No. I need to get two steps sidewards out of the office to see the windows of the appartment.
plan to add more remote clients? No just this one notebook.
The best connection I get is when I place all the WLAN Hardware within the office. The DI-624 is placed in a little room where alle the telefon and network stuff ist located. It has the original antenna. The Extender is placed together with the antenna on the closed place to the Appartment.
Allnet: HG2424G-NF ALLNET Antenne HyperGain 24dBi 24dBi Yagi
1,5m thin one.
No I cannot find this information.
I already returnd the extender once and I tried a Access Point with an Extender function. All the same problems.
I also think this may a explanation. Thanks Ciao Heiko
I cannot find thi scheckbox.
Thanks Ciao Heiko