Do you think the channel is the problem for the slow connection in the local network? Why's the communication to the internet is fine?
If I'll change the channel to 1, do you think it will solve the problem?
No your problem is the number or other wifi networks around you. Changing the channel MIGHT help a little but you still have a ton of other networks fighting you for the same space. You would be better going with a router that support 5Ghz 802.11n or a as it will most likley be less congested. If your tool above will scan the 5GHz space you should check it out to see how congested it is.
And also, why when I'm trying to ping the server by it's machine-name it says it can't find it, but when I'm pinging by it's IP and THEN ping it using the machine's-name, it works?
Sounds like your computer or default gateway is timing out when trying to resolve hostname to IP/MAC address during the ping. When you ping its IP it removes that extra step and can get a reply within the timeout period of the ping. Your system and the gateway then cache the destination MAC address based on the IP so the next time you ping the server it dosent have to re-discover the servers MAC.
Your wifi congestion could be the cause of this...
So you have no way to physically connect the two to the router temporarily to test?