Like I had warned, it is likely configuration, or as another person stated, the wire and port connectivity. IN the case of either, a new unit would not fix the issue.
Generally, system specs these days should not matter overly much on a semi-modern unit as all meet specs for a high speed internet connection.
Port and wire connectivity is fairly standard, I find it unlikely 4 different cables would not work, but I do not know condition of your cables either... so below is a simple enough guideline for pretty much anyone. I know from prior posts the unit itself is 10/100/1000, but I don't know the port using on the computer.
0. As stupid as it sounds, under no circumstances use phone wire for ethernet. Some connectors actually allow you to plug a 2-4 wire rj11/rj12 connector and phone wire into a 8 wire rj45 network ethernet port. Never do it, it will almost certainly not work right. I figure about the only way this could have occurred is if your unit came with a phone wire and were not looking when plugged it in.
1. Is the cable a Cat5, Cat5e or Cat6 or unlikely higher (Cat6+ or 7)? Minimum needed for 1000mhz/Gigabit Ethernet is a good Cat5e. Minimum needed for for 100mhz/Fast Ethernet is good Cat5 or any Cat6. Anything lower (Cat3, Cat4, bad or early gen Cat5) is 10mhz/Ethernet only and these days considered junk for networking. Also, make sure *all 8* wires are correctly used by that cable. Old network wires in a lot of cases used 4 of the 8. There are a *lot* of standards and terms used in network connectors, but just calling it by the Cat# standard it complies with normally is enough.
2. Do not mismatch port and wire... Do not put a Cat6 cable on a computer port that is rated for only 100mhz/Fast Ethernet.
3. Examine the boot or connector shielding, make sure it is completely intact and good condition. Badly shielded wires can cause vary odd things. I once had a badly shielded phone wire connected to a v.52 modem, the result, the modem speaker was playing radio stations as the bad wire was acting like an antenna and feeding back into the system.
I doubt you made your own wires, so I'm not going to go into wire patterns or crimp techniques.
If it is configuration...
1. I should note by default a modem/router/gateway is unlikely to work out of box without configuing unless you got it directly from the ISP. Look closely at the specs for that modem/router/gateway... then actually look on-line to see what your ISP actually requires... not all DSL modems work with all DSL ISP's. there is for example ADSL and VDSL, then there are the actual settings... some units have a wizard walkthrough and a large number of ISP configuations from a list.
2. If no wizard or on-line page you can see the settings your ISP uses, all is not lost. If you still have the old unit you used to have that worked with the ISP, even if it is broke, if you can access the settings on it you can compare them and see what might be different.
3. I do not know if you had any other devices connected during your trials, but try running it with a single cable to your system and no other systems/devices connected, it is possible some other device is interferring. Even if your *system* is all set up correctly and a good wire, if you had for example another system or a printer with a bad wire connected that could bring it down.
4. try setting your system to a static IP... I know that was mentioned before, but when doing so, also check your unit to insure #1, that it is not using that same IP, and #2, that the settings for the allowable IP range the unit allows is set to accept that IP. If not, then you will be unable to connect.
EDIT #5. Try networking, instead of internet. Like have a 2nd system connected by cable and try actually accessing it remotely and logging in and either copying some files or playing music or video form it, I'm wondering if this cutting out is also affecting networking, not just internet. It could give a clue as to where the problem is.
Outside of that... not too much I can say that can help you specifically.