Many manufacturers currently offer kitchen knives with Damascus steel blades at a premium.
Besides looking cool do such knifes have any better characteristics compared to simply good forged stainless steel kitchen knives?
At the time of my engineering degree (mid-1990s), the knowledge for true Damascus steel was lost, much like the knowledge of the recipes for the concrete used in the Roman Colosseum.
It's possible that more materials analysis has been performed since that point, as there have been a number of groups who would like to reproduce the process to determine how it compares to modern steel. (museums aren't willing to subject the known pieces to destructive testing).
With modern steel, they intentionally introduce small proportions of other metals to interrupt the crystals that form as the metal is cooled; this helps to improve the strength of the steel as there isn't a single sheer plane that could allow fractures through the whole item. They're also better able to control the cooling process, so that they can control what crystal structure forms in the steel as it's annealed and quenched.
My understanding of true Damascus steel is that it's likely two different crystaline structures, one more ductile (so it can compress to absorb more energy without failing) and the other more brittle (which can hold a sharper edge). The two work together similar to today's composite materials.
Most of the stuff sold on the market today is laminated from two or more metals. In many ways, it's more similar to the folding process of high-quality Japanese blades, but with dissimilar metals. As the sheets are mostly parallel to each other, I would suspect that the strength improvement isn't as high as the more erratic patterns in true Damascus, but this is likely countered by using metals that are independently stronger.
So, to answer the specific questions:
Most people are better off getting some decent but cheap knives and replacing them more often. Victorinox Fibrox regularly wins America's Test Kitchen's ratings of knives.
Firstly, true Damascus steel is a historical artifact--I infer you are talking about knives created from steel produced by reproduction methods which are similar.
This is a subjective question--only you can deside whether you find the value proposition favorable. The thing is, the qualities that make (reproduction) Damascus steel special and important (other than aesthetics) are perhaps more important to weaponry rather than culinary application.
The two main charactaristics that come to mind for Damascus steel, in my personal opinion, are:
Cutlery is not subjected to the same stress as weaponry, where being able to absorb impact stress without shattering is a relevant virtue.
But they are pretty.
My personal subjective opinion: no, not a good value proposition compared to good quality modern knives--either forged or stamped--like Shun, Wustof, Chicago Cutlery, Victorinox, or countless others.
As a custom knife builder my answer is simple: Yes and no. Some damascus blades are cheaply made and consist of simple layered steel or flattened steel cable that is etched to produce the lines that many people find attractive. They are definitely not worth any extra cost and in fact are easily outperformed by regular modern kitchen knives of moderate to good quality. In fact since these damascus blades are only formed from ordinary steel they will rust quite easily, must be maintained regularly and will not hold a keen edge.
Other types of damascus steel is made from layers of stainless steel (that results in a much more subtle pattern) but also has an inner core of high carbon specialty steel like VG-10. This type of damascus is far more expensive of course but produces a blade of superior quality as well as one that looks very cool. It will hold its edge longer because of the hardness of the inner core but will also be very durable due to the layers of more flexible and stain resistant stainless steel.
Whether or not it is worth the extra cost is entirely a personal choice. If you only use your knife to cut up hot dogs and store it in a drawer with the rest of your kitchen utensils, I'd say no. But if you use your knife extensively, like the very best and want a knife that looks (and stays) really sharp, I would say the answer is yes. But do your homework and ask questions. "Damascus" does not automatically mean better.
If you like the way a Damascus blade looks, it can be worth it. If you like a certain manufacturer and their high-end knives are damascus-clad, it can make sense to get those instead of their cheaper lines.
A modern "damascus" knife is simply two sheets of laminated damascus wallpaper around a hard, knife-steel core*. This core layer is going to determine how the knife works, how it cuts. So will the shape of the blade, the angle of the grind, shape of the handle, the weight distribution. Damascus cladding will not, it only changes the look of the knife, and it drives the price up a bit.
*Cladding a hard core with softer steel is a japanese invention. It enabled them to make swords with extremely sharp edges due to the hard core, without making the blade brittle. Soft steel is flexible, allowing the blade to flex without breaking. Damascus cladding has no functional differences to soft steel cladding. There are no downsides to it, other than the price.
Damascus steel offers no benefits over other types of good quality steel. They look cool, but don't do anything that other knives can't do much cheaper.