Let's see. 1st pic looks like 2 yoke connectors. I assume one comes directly off your tube's yoke around the neck. The other... is it permanently attached to your chassis, or just a pigtail included with the chassis? Likely the one from new chassis is not needed, as it appears that in another pic, you have 2 yoke connections on the new chassis (side by side). You should be able to just plug the tube's yoke connector on one of the 2 yoke connections on the new chassis without hacking wires. (The 2 connections might allow flipping without removing connector pins if the image ends up flipped/upside down. The Red/Blue SHOULD be for horizontal, and (if you look CLOSELY) you'll notice the Red/Blue are SLIGHTLY wider apart (on both connectors) than the Yellow/Brown (or Yellow/Green). So this keying should keep you ok, but I would definitely look at any documentation to make sure you're aligning the 2 horizontal coil wires with the correct pins on the chassis connectors, and respectively, the 2 vertical wires with the correct pins. If you accidentally flip EITHER horizontal OR vertical wires, it'll just have a flipped image. Just don't connect horizontal wires to vertical chassis connector pins and vice/versa.
Pic 2:
I assume the 2-part connector comes from your game board. Is the other permanently connected to the chassis or an included pigtail to mate with it's video signal connection on the new chassis? You're right, you can cut/strip these to make the new single connector plug right onto the chassis, but it'd be nice not to hack the cabinet wiring if possible. I'm sure you can make, or possibly buy, a converter pigtail. Just note Green mates to the White w/Green stripes it appears. The others look identical.
Pic 3:
This looks like it comes from the degauss coil on your tube (coil around tube covered in thick white insulation). Following your link early in this post, I looked at the new chassis. It clearly says it has NO degauss circuit and you'll have to use a manual degaussing coil/wand. So this connector is not used at all with the new chassis.
Pic 4:
They gave you a new dag wire assembly. I would probably cut the green wire off, solder it to the loose wire you cut (in Pic 6), if that is indeed the ground you cut that is connected to the old dag wire installed on your tube. Then it plugs likely onto the neckboard. The braided wire with the spring is not needed, as one should be on your tube (tough to see on the last blurry pic).
Pic 5:
No clue about the 2 red wires/connectors? Hard to see, but one has a +/- by the connector. Tube size selection maybe? Horizontal shift?