So "thank you" is תודה and the verb "to thank" (in this binyan) is מודה. Would that make the original root something like ידה?
No exactly, it comes from the same root as הודאה (confessing) and הודיה (something like "to praise (god)"). The root is י-ד-י (but many times י switched with ה or א)
So instead of saying tod or toda raba each and every time, can i also say: i thank you? Ani modeh/modah lekha/lakh?
Can this also be translated as "She is grateful to us"? That's how I have understood it when I heard it before.
Not directly, as מודה is an action and not a state. But I guess in some context it would mean the same thing.
Confess in hebrew is הודאה, when someone confessing to someone he (or she) is מתוודה בפני ("confess infront of somone's face"). Also מודה and מתוודה are a bit different. מודה (besides of thanking) mean to confirm an action or feeling in general, while מתוודה is to confirm bad-action and guilt, and it usually means that the person who התוודה is asking for forgivness. We even have a phrase about that "אני מודה ומתוודה ש..."