tl,dr: If your question is about money: Researchers in industry have significantly higher earning potential than researchers in academia. In my experience, industry is also a lot more personally rewarding, but that depends on what you are after.
So I want to know if or how a research scientist in industry get rewards to solve a problem or innovate while working on a project?
That's your day to day job. You get paid for it and your career advancement depends on your ability to do so. Your compensation is dependent on your abilty to innovate and typically higher (Sometimes a lot) as in academia
What does happen if they invent or innovate something which can be patented?
Depends a bit on local legislation, but any IP you generate (patent or trade secret) belongs generally to the company.
Do they get a share of potential financial benefits? Will be the research scientist part of the owner of the patent alongside of the company?
Again depends on the company, but typically not directly. You are still the "inventor" but you have no financial rights to the invention. Howver technical leaders and innovators will often be rewarded with equitiy or bonusses that are tied to the business success of the company. If your innovaton actively contribute to the business success this can be substantial.
HOWEVER, I think your question are missing the key points. I've been a researcher in both areas and personally found industry to be much more rewarding.
- Impact The vast majority of academic results in papers that are read only by a small handful of people working on the same stuff. There are about 100 million papers published per year and 99.99% of those are utterly inconsequential. So why bother ? In industry you have a chance to develop technologies that might make it into real products and have real impact on real people's live. I found it extremely rewarding if I see people happily using product that I help creating.
- Resourcing and priorities : Academia felt like an endless slog of writing grants and proposal that are often skewed to the current political funding prioprities and not what's needed or meaningful. Industry is business orientated, so at least the priorities are clear and easy to understand. Getting decent tools and equipment is much easier, since you can argue productivity (which doesn't matter in academia)
- Career Advancement : Academia has two or three steps up the ladder which tend to be very constrained and that's it. Industry offers a wide variety of roles where you can move between fundamental research, applied research, hands-on product development and various individual constributor and/or management/leadership paths.