There are 3 ways to solder the surface mount components:

Soldering Iron - Put some flux on the surface mount pads for the component you're working on. Melt a small amount of solder directly on a clean, small tip. Hold the component down on the board with tweezers while touching the soldering iron tip to each leg/pad long enough for the solder to spread. For the IC's and FPC connectors with multiple legs, slide the iron tip from leg to leg. Add more solder to the tip as needed.

Hot Air Rework Station - Put some solder paste on the PCB pads. Place the components on the pads.  Direct the hot air on the parts long enough to melt the paste. Try to blow the air directly down on the board (not at an angle). Reduce the airflow if the parts are not staying in place. 

Reflow Oven - Put solder paste on all the surface mount pads and then place each SMD component on the pads. Place the circuit board in the reflow oven and program the oven to the appropriate temperature profile for your solder paste.

Do a visual inspection with a magnifying glass, looking for any solder problems such as IC legs that are shorted together or pads that did not get enough solder. Use an ohm meter with stick pin probes that are small enough to allow you to check for shorts on the adjacent legs of all the IC’s and FPC connectors. Verify there are no power to ground shorts. Fix any shorts by applying flux and using solder wick. You may need to add back a little solder if the wick removed too much.

Use a Teensy 3.2 with header pins or solder your own header pins taking care to get them perfectly straight. The surface mount connections on the back side of the Teensy can be done with flying leads but a 2 x 7 SMD header is a lot easier to install. You'll need a long, skinny tip so your soldering iron can reach the pads. The alignment of the header is difficult to eye-ball. Use a small prototyping board with standard 0.1” hole spacing that can keep the header in position while you solder the pads on each end. You can use header pins or wires to attach Teensy pads AREF, A10, A11, A14, and GND2. The other Teensy pads are not used but can be connected to the board if you want (it looks better).

Solder both 10K potentiometers to the board (in the same orientation).

Verify the ohm meter shows no power to ground shorts for VIN, 3VA, and AREF. Connect the 3 FPC keyboard cables to the board. Connect a USB cable from the Teensy to your computer and verify the power pins are at 5V, 3.3V, and 2V. Adjust potentiometer R9 until the voltage on Teensy pin A11 is 1.0V. Adjust potentiometer R17 until the voltage on Teensy pin A10 is 1.0V

Compile and load the Teensy 3.2 code "IBM_380.ino"

Verify the trackpoint is functional as well as the left and right buttons. The trackpoint should not need excessive force to move the cursor. The cursor should not drift when the trackpoint is at rest.

Verify all keys on the keyboard are working. Verify that when Num-Lock is enabled, the appropriate keys act as a number pad. Use a volt meter on the keyboard LED pads to confirm CAPS LOCK, NUM LOCK, and SCROLL LOCK signals go to 3.3 volts when enabled.

Verify Fn-F1, Fn-F2, and Fn-F3 control the speaker Mute, Volume Down, and Volume Up.

Verify that repeatedly pushing Fn-F9 reduces the noise zone which causes the trackpoint to be more sensitive (it will react with a light touch). 

Verify pushing Fn-F11 restores the noise zone to the default value which causes the trackpoint to act the same as it did at power up.

Verify that repeatedly pushing Fn-F10 increases the noise zone which causes the trackpoint to be less sensitive (it will need to be pushed harder to move the cursor).

To check if Fn-F12 saves the current trackpoint position as the "at-rest" value, do the following:

Unplug the USB cable from your computer. Push and hold the trackpoint (in any direction) while plugging in the USB cable. Release the trackpoint after a few seconds. Use the trackpoint or the computer's mouse to attempt to bring the cursor back to center but it will be difficult. Release the trackpoint and push Fn-F12. This should capture the "at-rest" value and allow you to use the trackpoint normally again.