Hi Debbie, welcome to the forum and you have asked a question that is simple in the asking but the answer can range from simple to complex to impossible.
Not sure how things are in Finland, but here in the US, in the big companies and Government Departments the Purchasing staff (bean-counters) will make the decision based on a very detailed specification written by the Engineers.
As the company size decreases the closer the answers (acceptance) of the product will move to the Engineers.
Also, the smaller the company the more likely the CEO will stick his oar in and muddy things up as well. Often they make decisions the Engineers have to live with. It is foolhardy, hiring and Engineer then over-riding their decision, but that's the way it is.
The safest thing for you that I can suggest, is to deal with the person who requested the interest from your company for whatever it is you are asked to produce. Going around them to an Engineer will almost always lead to disaster for any bid you submit. I know it sucks and it is stupid, but you are dealing with human nature and that is a weird and often incomprehensible muddy pond of gray-matter.
"Choosing the company?" Personally I start with the years in the business, especially if it is an expensive purchase. Next I would do some background legwork and ask for referrals to other customers and *not* just satisfied customers. No one is perfect and a 100% satisfaction rate is impossible. Don't care who or what you are making. If you are in business long enough, mistakes will be made. It is how those mistakes are handled and corrected that is *very* important.
To me, price is a ways down the list unless it is overly expensive or unbelievably low-cost. I would ignore both ends of that spectrum. If you are making an off-the-shelf item then fixed and guaranteed prices is a must. I will not buy from a company that does not have a fixed price list for ready-made stuff.
If you are making something specific to the customer, then your company needs to do some ballpark costings as I can assure you the requesting company will have done in order to budget for whatever it is you are to supply.
Can't speak for everyone, but I like to get a good "feeling" about the company I am OKing a major purchase for. I get that feeling based on honesty, openness and genuine commitment to the project when dealing with all members of that company that I get to meet. It's all a matter of trust and that is earned, never assigned.