[This post assumes you actually need hardware design. If you really just want a Raspberry Pi with some custom inputs, none of this applies.]

Yes, there are hardware design firms. Your software experience may or may not help at all, depending on the type of product you're going for.

Basically, you need to find a company that does this. This is pretty hard to do. "Hardware design" firms may or may not do it, "embedded systems consultants" may or may not do it... you just need to find some local ones and talk to them. Some do hardware layouts, some do board manufacturing, some do industrial design, some do RF, some do product management -- decide what you need, then find a company that can handle it. Or find a company that can help you decide what you need :)

One way to find these companies is to call the local representative for chip manufacturers (Atmel, Texas Instruments, Silicon Labs, etc) and ask if they can recommend a local consulting house that does product development. They know everyone in your area who buys ICs. They'll be more inclined to work with you if you find a specific one of their chips that suits your needs, and tell them you're thinking about using it.

"How much would it cost" is mostly unanswerable. The answer, though, is definitely a LOT more than you suspect. I'm assuming you've phrased the question correctly and you actually need hardware design, but if you really just need to throw an Arduino or Raspberry Pi at something it will be much cheaper. But if you need custom hardware design, the answer is a LOT. Nobody can give an estimate without knowing what you're doing, of course... a digital thermometer will be pretty cheap, a self-driving car will be quite expensive. But creating a hardware product usually requires 1+ hardware designers, 1+ embedded software developers, 1+ industrial designer, and possibly more... and since you don't own a company, you'll be paying them consultant rates. $100+/hour each. Plus the cost of raw materials, which is high in low volumes, plus the cost of manufacturing a board, which is very high in low volumes, plus the cost of doing it over and over again until it works.

How long: again, nobody can answer without knowing what you're doing... but longer than you think. It takes a long time to manufacture a board. They come back broken, always. It takes a while to figure out why they're broken, then you get to make the next revision. Rinse and repeat. Don't forget that you'll probably need something -- an IC, a JTAG debugger, a person -- that has a 6-month lead time. And it's easy to get stuck on a bug for a few weeks/months that turns out to be a defect in the microprocessor you're using. There are a lot of things that can stall the project and burn time, and nearly every project hits one.

The cost and time of projects is all over the board, and very hard to predict, but it's not unusual for a very "simple" project (ex: USB microphone) to run up $250,000 and 6 months for 10 prototypes, and a complex project (ex: television cable box) to run over $2 million and 1.5 years. On the other hand, maybe you luck out and your microphone works perfectly and only runs $30,000. Or nothing works, and you throw paycheck after paycheck at it, until you're down $5 million and years have passed and people don't even use USB anymore.

And that's just to get a prototype.

Something to remember: a digital thermometer costs $5 at every corner store because there are digital thermometer companies with a very large staff dedicated to making millions and millions of digital thermometers per year. They can make them for $0.50 each. You are not a digital thermometer company, and you will not make one for $0.50... your first one will cost $100,000. Hardware is only profitable if each unit is worth a fortune, or you're going to sell millions of units.

I strongly recommend it -- hardware is a blast -- but be aware that it is far, far, far more volatile and uncontrollable than software development.