By burned up I mean crispy critters! Wiring that is not sized properly for the load, or that chafes against metal and shorts out, or that suffers from a poor joint that eventually fails or overheats can cause a vehicle fire. I'm not suggesting that the job they did on yours will cause it, but the risk is increased with shoddy wiring like you have pictured there.
The purple wire to the relay is obviously the powered lead. The red and blue wires are each big enough that they could actually be carrying a small load. One of them is probably the signal (to the coil) and the other is the switched wire (to the accessory). Usually there is a fourth terminal on the relay to act as the ground path for the coil, but your signal wire might provide the ground. Then the purple would be jumped across two terminals to provide power to the coil AND the accessory circuit. Sometimes the ground is the metal case that needs to be screwed to a metal ground, but that was more common back in the '60's and '70's. With the '80's a lot of car brands had switched over to wired grounds for lights and other circuits so that they could have plastic housings instead of metal. Your relay appears to be plastic.
The wire that they tapped into the huge cable should be removed and a joint made where the cable ends at a connector, or run all the way to the source itself. The red wire that has been joined and taped simply needs to be soldered if it is not one solid piece already. The blue and purple wires that appear to be wired to the relay need to be gathered into loops and tied off. All of the wires added need to be checked to be sure that they aren't chafing on anything that will eventually rub through the insulation and cause a short. Look especially where they go through holes. At the very least a fuse will blow and you'll lose the function of any circuit on that branch. At worst your car will catch fire.
I don't say this to scare you (or any others that read this), but to make you aware of just how serious it can be. If you take reasonable precautions you should have an event-free experience from your mods.
The range of your remote is mostly determined by the location of the receiving antenna. The transmitter is the fob. If the receiver is behind a metal bulkhead, such as being hidden in the dash or down in the kick panel, then it will have difficulty receiving the fob's transmission. Placing it up high, like on top of the dash, attached to a window, or up in the headliner (with the antenna properly oriented) can greatly increase the effective range of the fob. Also look at the battery terminals inside the fob. If they are flopping around they might need to be resoldered. If they don't grip the battery (batteries) firmly then they may need to be bent a little to grip better.
Post up what steps you take to make your car better, okay?