Outlet turning on ceiling fan

"The bridge between the screws was not broken" Are you 100% sure of that? Because the common use for black and red wires and a two-pole breaker in a kitchen outlet circuit is for a common neutral circuit (multiwire branch circuit), and shorting the two phases together with the new outlet would produce exactly the symptoms you have seen. So I'm going to suggest that the two sides of the old outlet are not actually connected, and if you don't have a meter you can check that with, I'll provide you an alternate test that does not involve looking at the connection, which you've already done and come to (I suspect) the wrong conclusion about:

Cap the red (or black) wire and connect only the black (or red) wire to the old outlet. Plug something into one receptacle, and see if has power. Then plug it into the other side. Repeat with the other wire (only) connected to the other side (only.) If I'm right, in each case, only half the outlet will have power, and which half will change with which screw you connect to.

It appears that the device you have probably only has one hot screw, so it cannot be connected the same way as the old outlet with two separate hots; so leave one wire capped off, or install it somewhere other than your kitchen (the reason for using this type of circuit in the kitchen is to provide more circuits to the countertop outlets - this device defeats that if it cannot have two separate hot feeds.)

Yes, this should work. A few things you may have already considered:

1. Each time your cable goes into or out of a junction box you should plan to lose 6" in length.

   So tying into and out of the ceiling box for the fan would require 12" of slack on the existing 4-wire cable. Your plan to add a separate junction box is a better approach. The trick is to put it at least 6" closer to the dual-wired receptacle and run 3-wire the longer distance to this box. When you tie into the 4-wire here you will have extra 4-wire to use in the ceiling fan box.

2. Don't disconnect the red wire going to the duplex receptacle.

   Just tie red and black to the un-switched black in your junction box. This puts half the load on each wire and you don't have to rewire the receptacle in the wall.

3. Get a deep 2-gang switch box to fit all those connectors

   and dissipate the heat any dimmers you use to control fan speed or dim the fan lights. You're looking at needing at more than 29 cubic inches not counting extra for dimmers. And you always want to leave room for future expansion.

4. Get a ceiling box designed to hold a fan and mount it securely to a joist.