I was thinking of getting a multimeter with a temperature probe.
Do not bother if it is to check soldering iron temperature. It will not work.
You will need to solder the thermocouple to get the temperature but then you have ruined the thermocouple.
If the iron melts solder, it is working.
Soldering is down to techique. It is also down to personnel preference.
Work out what works for you, stick to it unless someone shows you something better.
The stranded wire will eventually break if the circuit is handled or subject to vibration.
I have not had that problem.
I make sure that link wires on strip board is flat against the board and exactly the right length, ie no loose wires that will shake or move.
For very early learning, use what you have and keep the costs down. Develop your own techniques.
Hope this helps.