milk in cake baking. does age of standard whole milk matter a lot?

The quality of your milk is usually based off the fat-content. Unfortunately fat's oxidise and create rancid odours and flavours which are carried into baked products.
A good whole-fat milk should taste (faintly) of vanilla with a hint of caramel, and a deep, pleasent, buttery flavour.

All of which will work in your cake.

Protect your milk from light and air as best as you can, glass and plastic bottles worsten the quality more rapidly.

Eventually the lactose in the milk will encourage enough lactic-acid bacteria to acidify the milk, not a big problem, however will affect recipes with leavening...etc... the fats are what you need to protect for flavour.

Rancid smells will create rancid baked products.

in general what the milks smell/tastes like will affect the smell/taste of your baked product. Buttermilks add a depth and flavor that whole milk does not, goats milk, yaks milk, even yogurt. They are all used in breads resulting in all these wonderful flavors, though some, I am sure, also affect the rise and texture of the bread too. the chemistry of baking is an art I both admire and fear.