Yes. Maybe turn off/disable the WiFi radio on your older modem so that the new router isn’t impacted by the radio noise from the old modem.

If there is a radio spectrum issue or a deeper down networking issue, replacing the gear may not solve the problem. So I’d try to isolate the current setup’s issue before blindly replacing it.

A wired backhaul is always going to provide better performance, reliability, and consistency… as well as the lowest latency. Wireless backhaul is going to be more problematic in a radio-heavy area, or if your building has a lot of dense materials. But otherwise a wireless backhaul may be perfectly suitable for your needs.

Four nodes can be a lot. The most nodes I have installed is three, in homes that are about 4500 sq ft. These are in modern North American “wood & drywall” homes, and I was able to position the nodes well, so your situation may demand something else. Generally I want to use as few nodes as I can, because there is a system performance overhead for each node.

6 GHz is great if you have a lot of 6E gear, but it is worse than 5 GHz in terms of traversing walls and materials. So if you have an open floorplan it’s a good option, but otherwise I’d save my money. I only have one 6e kit installed in a home, and there has been at most one client I’ve seen connected to 6 GHz. The backhaul prefers 5 GHz.

You have a couple options. The easiest is to buy an adjustable power adapter. They’re about $20 on Amazon. They come with a bunch of “barrel tips” so it is very likely you’ll have one that matches. I have one of these adjustables to test and validate devices when the power adapter is missing or suspect. Pretty handy.

Otherwise, you need to know the voltage, polarity, minimum amperage, and jack size. The first three are easy because they are almost always declared on the old brick or the device itself. You don’t have to match the amperage, but the new adapter needs to supply at least the amperage of the original. For example, if your old supply was 1000 mA, a 1.5 Amp supply is fine but a 500 mA supply will not provide adequate power.

The physical jacks: you get to learn what they are through experience. The most popular are 2.1mm in diameter and 5.5 mm in length, with 2.5 mm diameter being the 2nd most popular. But there are many others. Sometimes if you look for a replacement adapter for your device the specs will actually say. Otherwise you’ll need to measure your old one.

If your home router is “dropping bands”, it is likely that the power adapter has weakened and can no longer deliver enough amps to the device to keep the radios operating properly, or not enough power to keep the CPU from losing bits and crashing (LEDs on, but not reliably functional).

I’d replace the power adapter with one that can deliver more amps. These adapters have to deal with every power transient and are often powered up for years. And they are often mediocre.