"Pour vous rendre service, je vais répondre à votre question."

"Pour ne pas vous décevoir, je vais jouer cette sonate."

In your example, I would rather write

"Quant à elle, je ne l’aime plus."

You can start your sentence with pour, but for your example, you cannot just say pour elle and have it mean as for her. That’s because in pour elle, the object will be understood as the beneficiary of some action. It would either mean something like for her own good or from her point of view rather than in regards to her.

If you really want to start with pour, you can use pour ce qui, which will help avoid the misunderstanding:

Pour ce qui la concerne, …

Pour ce qui est de cette femme, …

Pour elle, je ferais n’importe quoi is correct.

Pour elle, je ne l’aime plus doesn’t have the meaning you want in french (in another context, that could mean that she thinks you don’t love her anymore).

Google’s version is more accurate. But as Bernard Massé suggested, Quant à elle works too.

Sentences originally adopted from: “A French Reference Grammar” by H. Ferrar and slightly modified by me. But see also the remarks of @jlliagre in the associated comments.

Pour (among the many meanings) may express personal concern:

Pour ma part, je m’abstiendrai. Personally, I shall abstain.

It may start a phrase when it also expresses:

1. By way of:

Pour couverture il avait un sac. For a blanket he had a sack.
(As @jlliagre suggests, it is better here to use comme couverture.)

2. Considering

Pour son âge/pour un enfant, il lit bien.

3. Purpose

Pour le terminer, il faut s’entraîner.

Pour n’être pas vu, il nous faut sortir la nuit.
(As @jlliagre remarks it is better to use pour ne pas être vu instead.)