To get started on the Internet, should we first focus on the product or the company? Can we create a product without having company? If Yes, it is an handicap?
Charles the very first step in any successful venture is solving a problem. This creates the entire framework on the steps needed. Can the problem be solved with a product? If so create the product. If the problem requires a solution that requires collaboration then collaborate. All great companies started this way. Google solved a better and faster way to search. Facebook started a way to meet people. Companies and industries that do not solve problems soon fail. It is the most basic and useful tool there is available. Who has this problem? (your market) How big of a problem is it? (competition) What are the issues this problem is causing? (how much will they pay) In a startup idea phase I think it is best to solve a problem you are having or have had in past experiences. The focus should be not on product or company it is problem research. Hope this helps.
You are doing both.
You can start working on the product ASAP. But you have to be prepared to do the company stuff. If you (and your co-founders) start to ask yourselves "should we do x", then you should be doing x. For instance, if there are multiple founders, you should have a founders agreement written up (even if it's basic) before you work on product. What if one of you contributes more or less during the product part? What if one has to go get a real job down the line? You'll have already laid out who owns what and what happens in case of a split. Last thing you want is that founder coming back later and demanding their third of the company because you guys didn't do company stuff early enough.
There are certain things that should be done as early as possible. Not having the company stuff done can bite you in the ass later on. Part of running a startup is being able to do both. The other part of running a startup (and the one that plays a big role in you remaining in charge during any funding) is transitioning into being a company. If you have already been doing the "company" basics...it shows you are more prepared than the average startup entrepreneur to transition from startup to company.
Making a product is not a goal, it is a stepping stone. Being a company is the goal.
You are selling a product, not a company. Start with the product. The company is just a conduit to get things done (legally and logistically). The product, market and people should be your focus.
Neither. You should start with the customer. It is more difficult to identify the right problem to solve and the right customer to target than to create a product or create the vehicle (company). What you think you know about both, problem and customer, is many times a mirage based on good faith assumptions that you need to validate with hard data. With enlisted willing customers, figuring out the product (MVP) is easier, and the legal paperwork an afterthought.
A startup is a temporary construct, not an operating company, whose job it is to define the problem, solution, customer, desired market and industry, basically find a way to make your solution / idea feasible (technically, financially, market) in the chosen marketplace. You will identify a group of paying customers whom you think you can serve.
Once you are comfortable with and have mitigated as many risks as you can find, you can then step off, start a company and run like the wind.
This process is not a handicap. In fact the research you will do to discover all that feasibility stuff can prevent you from wasting a lot of money and time chasing something that won't ever work.
Are you talking about legally establishing a business (e.g., C-corp, LLC, etc.)? Typically not necessary as a first step. As others have mentioned, you can make a lot of progress speaking with potential customers and gaining a understanding the problem you're trying to solve, how important a problem it is to solve, how much they'll pay, understanding the market, etc. before forming a company. If you're working with co-founders, it's a good idea to have a simple partnership agreement and have NDAs/invention and IP assignments/non-competes/non-solicits from day one so as to avoid issues down the road. A tool I'm building (RiffBiz.com) walks entrepreneurs through the exact steps they should take while building their business. It's based on my interactions with hundreds of startups during my ~20 years in silicon valley as an entrepreneur, investor, advisor, and acquirer of tech businesses.