Electronic – Soldering breadboard wires onto a small chip

A complete answer to this question would be quite long. I'll give you some guidelines.

• Your serial flash may work under the SPI protocol. Most MCUs (including the one in your dev board) include hardware modules to communicate using SPI. If you use SPI to test your serial flash, you'll be able to benefit from the advantages that those modules provide: speed + robustness + less code to do the same + lower MCU utilization.
• The MCU in your dev board is an STM32F207IGH6, with UFBGA176 package. The datasheet for it, in pages 40 to 51, states correspondence between peripheral signals and pin numbers. The MCU has 3 SPI modules. Each SPI module involves 4 signals (NSS / SCK / MISO / MOSI). At least some subset of the signals of each one of them is already used in your dev board. This means that, to gain complete control of a full SPI bus, you need to remove some parts (like resistors), or open jumpers, in your dev board. Just because of this, and if the purpose is just to test serial flashes, I would recommend you to use another dev board. There are plenty that give you access to complete SPI buses.
• For instance, SPI1 is mapped to pins PA4, PA5, PA6 and PA7. Those are available at connector CN2 of your dev board. However, those signals are already used by some functionalities, in your dev board. If you want that SPI1 bus to connect your serial flash to the MCU, and nothing else, you need to remove R115, the camera module, R69, RS2 and open JP8.
• In the serial flash datasheet, read section "Serial Peripheral Interface Modes" and look at Figures 6 to 8, to see how you need to make the SPI connections. In fact, if you don't know much about SPI, best thing you could do first is to learn about it. It can't be explained in one paragraph.
• Yes, you can use GPIO pins, instead of an SPI port, but you will need to big-bang the protocol.

You should use a wire stripper or a pair of pliers to strip insulation from wire. Small wires can be stripped with your bare hands: just pinch the insulation with a fingernail and yank it off. If you catch it just right, the insulation tears and slides off, but the conductor is left intact.

It sounds like by burning the insulation, you have fouled the metal with the products of combustion of the plastic. This layer will prevent the solder from flowing onto the metal and bonding with it. Even if the plastic was completely vaporized, it's likely that the metal oxidized. Solder does not like oxidized surfaces. To solder, you need clean surfaces free of oxidation or other residue. (Other than soldering flux. Speaking of which, you should have some and use it!)

One of the functions of soldering flux is that, being mildly acidic, when it gets hot, it etches away the thin layer of oxidation from copper. It will probably not eat through the carbonated remains of burnt insulation, however. Still, it is worth a try. If flux doesn't cut through, then just cut a piece off the end of the wire and try again, and this time strip it mechanically rather than, err, pyrognostically. :)

Update: if the wire insulation is just a fine lacquer (similar to what is used on wire used for winding voice coils or transformers), it may be possible to solder with the lacquer in place, with better results than torching the lacquer. Some hobbyists spray lacquer on DIY circuit boards, without covering the solder pads, and solder right through it anyway. It may also be possible to remove the lacquer by using a solvent such as acetone.