Are professional ranks representative for how many stones of handicap players can give each other?
Historically, pro ranks were an indicator of playing strength. It was said that 3 (later: 4) ranks are about a stone difference. To my knowledge, there never was a time when 1 rank difference actually meant 1 stone.

In the 20th century, there was a sudden and increasing change in the strength of new pros. This is generally considered to be a consequence of intensified training in insei schools as well as the advent of the internet and the worldwide communication it enabled.

For instance, earlier, Go schools occasionally developed secret moves - new moves that would stump many opponents who were not familiar with it. Nowadays, such a move would be known by millions right after it is played, and days later, everyone would have studied it already. Modern insei schools make use of this and always present their students with the most up-to-date material and worldwide research, making sure their students are fully aware of the current moves in Go.

Also, there is incredibly fierce competition among the insei. Only a handful of new pros is selected each year, and even the slightest mistake will ruin your chance to become pro. Therefore, everyone is working extremely hard.

This leads to new pros being able to compete on par with previous pros. They already possess the same knowledge as their seniors, and thanks to their young age get less easily exhausted from long, intense games, they are sometimes even said to be stronger than older pros.

In fact, to be successful as a professional Go player, you have to become pro early (around the age 15) and win titles while you still have a slight edge over the competition. In just 10, or 15 max, years, you'll be overwhelmed by the next generation. Very few pros could stay at the top for an extended period of time, one of the most notable being Go Seigen, who dominated the Go world for 30 years. But apart from exceptions like him, modern pros, even huge talents like Lee Sedol, are falling behind very quickly.

This means that rank is basically meaningless. A new 1p may well as strong be as an older 9d. Or the other way around. I asked several pros about this and the reply was always the same: Rank is no longer a useful measurement of playing strength. Personally, I call it an 'experience indicator', but even that is not completely correct.

To compare pros, you need to look at their results in games over time. There are various lists that attempt precisely that, and those lists are a lot more useful than the rank given. Since young pros who do very well get promoted quickly they ensure that the top spots in those lists tend to be occupied by 9 dans, still.

Interestingly, high dan promotions in China and Korea are linked to great performance in title tournaments - both for amateurs and professionals. For instance, a Chinese 6 dan (amateur) is a 5 dan who won a regional tournament. A Chinese 8 dan (amateur) is the current champion of all China (meaning there is only one of them at any time), former champions are 7 dan (*). In a similar fashion, winning an international tournament yields an immediate promotion from 1p to 8p in Korea, while regularly, advancing the ranks is a lot slower. (I'm not 100% up-to-date on details of this topic though.)

Seen this way, ranks of new pros can be seen as a measurement for achievements. Being awarded 9p is indication of outstanding accomplishments. A rank of 1p should not be seen as failure, especially with young pros it simply means their accomplishments will be for later.

Edited to add in 2014:

At times there is a promotion for professionals that specifically promote Go in the West. This is meant to compensate for their lack of tournament experience that is implied by their increased teaching. Since pro rank is an important factor in payment (when teaching), this is an important help for pros working in the West. I am specifically aware of a Korean promotion of Korean professionals from 8p to 9p or from 1p-7p to 8p.

(*) This information about Chinese 6d-8d amateur players appears to be incorrect. I've received multiple explanations as to what the current state actually is, and they all differed, so I'm at a loss here.