is there a big difference between "tu" and "tady"?
no difference in the meaning
Is there an etymological connection between vyšetřovat to investigate and šetřit to save? (And if so, what's the connection?)
šetřiti (koho, čeho) in old Czech meant to save (someone, something from something), pay attention to something, care about something, look at something, watch something, take something into consideration