Learn Italian: Invecchiando, le rughe vanno aumentando. - Getting older, wrinkles are gradually increasing.

Stanno aumentando (stare + gerund) simply means the wrinkles are increasing right now. Using vanno aumentando (andare + gerund) emphasizes a gradual, step-by-step progression over a long period. It highlights that the wrinkles are accumulating little by little as time marches on.

How does the single word invecchiando mean "Getting older" without a word for "while" or "as"?

In Italian, a gerund on its own frequently acts as an entire background clause meaning "while doing something" or "as something happens." You don't need to add a conjunction like mentre (while) before it. Invecchiando neatly packs the concept of "as one ages" into a single word.

Grammatically, does invecchiando share the subject of the main clause (le rughe)? Are the wrinkles aging?

Strictly speaking, Italian grammar dictates that a gerund should share the subject of the main verb. However, just like in English, it is incredibly common in everyday Italian to use a gerund for a general, impersonal observation. Context makes it completely clear that the implicit subject of invecchiando is people in general, not the wrinkles.

If I wanted to change this to the past and say "having gotten older", what auxiliary verb would I use?