Hi again Derek, let's see...
1. investigar, inquirir, indagar;
Investigar and Indagar are very similar (investigate, research), indagar is less frequent, at least in Spain. Inquirir means to ask a question (which of course, can be part of an investigation).
2. arreglar, reparar;
These are the same. Reparar a bit more formal.
3. medida, talla;
Medida is "measure" and talla is "size". Medida would be in centimetres, inches, or whatever unit, and Talla would be S/M/L/XL, 30/32/34, or whatever the manufacturer decided to use.
4. estación, temporada;
Estación refers only to the four seasons of the year (estación invernal). Temporada can be any period of time (temporada de vacaciones, temporada otoño-invierno, temporada de vendimia).
5. ondulado, rizado( both of them could mean "wavy and curly hair")
They don´t mean "wave and curly", the same way that wavy doesn´t mean curly! Ondulado = wavy, and Rizado = curly. Ondulado/wavy is a curve shape that doesn´t close on itself, whereas Curly/Rizado is a closed (or almost closed) loop. To check the difference maybe look for google images of curly hair vs wavy hair, they´re very different!