"My personal opinion is..." Is it always pointless to use the words "personal" and "personally"?

Generally, the word personal is used in these scenarios to indicate (or just emphasize) that the matter is, in fact, personal (from themselves without any other context to affect it). Let's take a look at your example sentences.

She is a personal friend of mine.

Without the "personal" there, there's no telling exactly how they're friends. A friend from work? A friend from school? Calling the friend a personal friend implies a somewhat less superficial relationship.

It's my personal opinion...

A person can have multiple opinions on a matter. For example, a climate scientist's professional opinion on whether global warning is an anthropogenic phenomenon would probably be that the evidence points towards it being so, while personally they may think it's just the perpetuation of natural climate cycles.

And as some others have pointed out, a public figure might prefix her opinion with "personal" when talking publicly in order to separate her own opinion from the opinion of her company or organization.

Personally, I would advise you...

A lawyer might give his friend who isn't a client some off-the-record advice on what will happen should they take a lawsuit or proceeding. A police officer might give a person he's talking to some unofficial guidance as to how to deal with a charge he's filing.

The manager said he will examine the matter personally.

If the manager were not examining the matter personally, he might examine it via one of his subordinates who would give him a report of the matter.

Personally, I don't care whether...

Again, a person may be forced to assume a particular impersonal opinion by virtue of their occupation. A traffic officer might not personally care that somebody's driving 20 km/h above the speed limit on a highway when everybody else is 15 over, but professionally he could never say that.

I have a personal interest in the matter.

If you advocate against pirating a game developer's games because you have stock investments in their company and don't want to see your investment lost, that's a business interest. If you advocate against pirating those games just because you think piracy is wrong and people shouldn't be stealing, that's a moral interest. But if you advocate against pirating those games because you like that developer's games and want to continue to have more games from that developer to play, that's a personal interest.

So yes, the word personal is indeed useful in these contexts, and not completely redundant either.

Personal is being used in at least three non-redundant ways in these examples

• To distinguish a personal opinion from another opinion, for example a professional opinion, given by a professional person. It's my personal opinion....

• As emphasis that the opinion is mine, and may differ from others. Personally, I would advise you...

• To express a connection with the matter I have a personal interest in the matter.

The manager will examine the matter personally, rather than assign it to a subordinate.

The only case that appears redundant is She is a personal friend of mine.