LAN keeps disconnecting and connecting constantly

Funny thing happened this evening. I was in a rush and haven't had time to reply, so I left my PC in sleep mode. I decided to post a response when I get back. A couple of hours later when I got back I noticed that the computer is „not connected“. I tried to check Network and Sharing center but I wasn't able to open it at all. Obviously the connection was completely lost while the computer has been „on sleep“. This was the first time such thing happened. I am writing this on my laptop.
In regard of your first question, the answer is yes.
Since I can not open Network and Sharing center at the moment, I am not able to post a screen of the DHCP setting, but I run the ipconfig /all command. I copied it to USB stick and here it is:
C:\Windows\system32>ipconfig /all
Windows IP Configuration
Host Name . . . . . . . . . . . . : xxxxxe-PC
Primary Dns Suffix . . . . . . . :
Node Type . . . . . . . . . . . . : Hybrid
IP Routing Enabled. . . . . . . . : No
WINS Proxy Enabled. . . . . . . . : No
Ethernet adapter Local Area Connection 2:
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Marvell Yukon 88E8056 PCI-E Gigabit Ethernet Controller
Physical Address. . . . . . . . . : 00-1D-60-C9-83-46
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
Link-local IPv6 Address . . . . . : fe80::18ba:d261:218a:78e7%16(Preferred)
IPv4 Address. . . . . . . . . . . : 192.168.1.20(Preferred)
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.1.1
DHCPv6 IAID . . . . . . . . . . . : 469769568
DHCPv6 Client DUID. . . . . . . . : 00-01-00-01-1C-7C-EA-F5-00-1D-60-C9-83-46
DNS Servers . . . . . . . . . . . : 85.114.32.7
85.114.32.8
NetBIOS over Tcpip. . . . . . . . : Enabled
Ethernet adapter Local Area Connection:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Realtek PCI GBE Family Controller
Physical Address. . . . . . . . . : 00-1D-60-C9-78-7A
DHCP Enabled. . . . . . . . . . . : Yes
Autoconfiguration Enabled . . . . : Yes
Wireless LAN adapter Wireless Network Connection:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Realtek RTL8187 Wireless 802.11b/g 54Mbps
USB 2.0 Network Adapter
Physical Address. . . . . . . . . : 00-15-AF-3C-86-E8
DHCP Enabled. . . . . . . . . . . : Yes
Autoconfiguration Enabled . . . . : Yes
Tunnel adapter Local Area Connection* 9:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Teredo Tunneling Pseudo-Interface
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
Tunnel adapter isatap.{687D226F-59BA-4FB6-ACB8-DB0B5785461B}:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Microsoft ISATAP Adapter
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
Tunnel adapter isatap.{56944A06-2190-41C5-8A28-1F82B8549D3C}:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Microsoft ISATAP Adapter #3
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
Tunnel adapter isatap.{37ACE7DF-869D-4B46-BC91-498AE2D4AF31}:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Microsoft ISATAP Adapter #4
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
C:\Windows\system32>

Well, he has the Realtek disabled, but I am thinking it's just a conflict with the DHCP range of the router.

OP - open your router's DHCP settings and post the DHCP range when you have time.

By checking DHCP range, do you mean this? I got these numbers from my ISP provider and I entered those manually. When I check "Obtain an IP address automatically" nothing happens. The fields remain blank.

Nope. You need to open your browser and go to 192.168.1.1
That will open your router's configuration page; look for DHCP range, which is usually on the first page,but you may have to look around a bit, depending on the router brand/model.

It would also be a good idea to change your router's default password at that time.

This is like a sitcom. Another complication popped up.
I opened FF (and I have also tried with IE later) and went to 192.168.1.1
Authentication was required. I called my ISP and they gave me username and password. Which DIDN't WORK. I tried zillion times (small caps and all). Then I connected my laptop to router and tried the same procedure again. Username and password were not accepted. But the ISP agent told me that the DHCP range should be 0-255.
I must stress that there is (and always was) only one PC connected to the router.

Are you plugging in directly to the ISP modem or do you have a router that connects to the modem?

Either way, you can google its name (router or modem) with "default login" and should be able to find the user name/password. For example, if you are using a Linksys WRT54GL router, you can google "Linksys WRT54GL default login", etc.

We need the DHCP info, since most routers don't reserve over the entire network. The DHCP info should be something like the red square on this picture:

The modem is directly connected to PC via cable.
The default pass for my modem does not work for some reason, though I haven't changed it since the installation. (I don't see your attachment,btw)

You should be bale to reset the modem with a hardware button - that would reverse the password to the default.