No, most commercial carpet is installed by glueing it down to bare concrete without any underpad under it. But, that's typically done only in commercial settings where perhaps part of the reason is that carpet without underpad offers less roll resistance for people in wheelchairs. Also, people tend to drag their feet when they walk, so they're more inclined to catch their foot on a softer carpet with underpad and trip than on a harder carpet without a pad.
In a residential setting, virtually all carpet installed in private homes is installed over underpad. The underpad makes the carpet feel warmer if it's installed over concrete or you're walking on it with socks or bare feet, it makes the carpet last longer by absorbing much of the shock of each foot fall, and it, and it makes thin pile carpets feel thicker. If I were you, I would definitely install your carpet over 6 to 8 pound foam chip underpad. That means the foam chip pad weighs 6 to 8 pounds per cubic foot. That's the underpad that gives you the most bang for the buck, and it's probably the most popular underpad for residential carpets.
That MAY be a problem, depending on how much you're willing to pay for the installation of the carpet. You see, carpet comes in three different widths; 12 foot wide, 13' 6" wide and 15 foot wide, with 12' being the most common by far. About the only carpets that come 15 feet wide are Berbers, and so far as I know, solution dyed nylon carpet only comes in level loop commercial carpet, not in any berbers.
So, if you want to save on installation costs, you can opt to have a 15 foot wide berber installed, but you won't find one made with solution dyed nylon. You will find ordinary dyed nylon 15 foot wide berbers and 100% Olefin (pronounced: "polypropylene") 15 foot wide berbers. The ordinary nylon carpets will be dyed normally, and so you can't use bleach on them without taking the colour out. The 100% Olefin carpets will be solution dyed, so you can use bleach without affecting the colour, but Olefin is a weaker fiber and the carpet tends to show wear patterns a lot sooner.
You can buy a 12 foot wide carpet in a solution dyed nylon, but carpet has a "grain" to it (kinda like wood) in that each tuft of the carpet will be leaning slightly in the direction it came off the carpet mill. That lean is permanent.
If you want to install a 12 foot wide carpet in a 15 foot wide room, you have to ensure that the tufts are all leaning in the same direction everywhere in the room. So, in that case you'll have to order a 18 foot 9 inch long piece of 12 foot wide carpet and cut a 12 foot wide, 3' 9" inch long piece off the end. Then, you cut that 12 foot wide, 3 foot 9 inch long piece into four pieces each 3 feet wide and 3 feet 9 inches long. The installers will hot melt tape those pieces end-to-end, being careful that the tufts lean in the same direction on each piece, to make a strip of carpet 3 feet wide and 15 feet long. They will then hot melt tape that to the 15 foot long by 12 foot wide piece of carpet to make a carpet 15 feet wide and 15 feet long that they can install into your room.
Obviously, it takes more time and labour to do that than to install a berber carpet that comes 15 feet wide.
Will hardware stores like Lowes or Home Depot have this type of carpet?
I kinda doubt that Lowes or Home Depot would sell either solution dyed nylon carpet or 15 foot wide carpet, but you should ask. They will sell 12 foot wide 100% Olefin carpet, tho, as well as 12 foot wide conventionally dyed nylon carpet. There's no harm in asking. Maybe print off this post and explain the problem to the flooring salesman there, and see if they'll special order carpet for you.
I am not experienced in buying carpet, but will I get a better price from a smaller retailer or should i stick with Lowes/Home Depot?
I'd say you're sure to get a better price on the carpet of your choice from a home center simply because their business model is to sell in high volume at lower prices. You get better service and more knowledgeable sales staff at a privately owned carpet retail store.
