A couple of things...

Question

pour que je choisisse quelque chose à lire avec elle - why is the "à" used? I can't find an expression with "choisir à"?
 
Au cours des années, elle m'a transmis sa passion - why is passé composé used? She clearly says 'over the years' so a repeated action? surely the imparfait would be used here?

Accepted Answer

The construction is: quelque chose à + [infinitive]

Answer

Bonjour James,
- Chris is correct that the preposition "à" is linked to "quelque chose".
- Le Passé Composé with "transmettre" is used as the action happened once in this time frame. 
Au cours des années, elle me transmettait sa passion
 -> she kept on passing on a passion
It isn't the case here as it is a result that happened once through that established time frame. Therefore, it is a single completed action.
I hope this is helpful.
Bonne journée !