It is definitely OK to construct and use your own test as long as the material (e.g., items, questions, problems) does not represent a direct copy of existing copy-righted test material. However, in that case you would have to run your own test validation (and other) studies which may be very costly and time-consuming (rather large samples and potentially complex statistical analyses are usually required to examine reliability, validity and other psychometric properties of tests). In addition, you may have to revise the test (and run additional studies) if you find that some of the psychometric properties of the initial test version are not acceptable.

These are some of the reasons why most tests are not free and rather expensive. A lot of money goes into developing tests, validating them, collecting and analyzing norm samples, etc. It's a quite complicated endeavor...

It is definitely possible, and some researchers believe that it is ethically preferrable to do so. In 1999 I had a chat with the late Robbie Case, who had invented valid, reliable, and useful working memory tests, such as the Counting Span and the Mr.Cucumber Test. He told me that he did not copyright them because he wanted everybody to be free to use them at no cost.

I think that this is highly commendable; however, it also has two disadvantages. First, several researchers thought that they could improve these tests or manipulate them for experimental purposes, so now there is no "standard" version of the Counting Span or the Mr.Cucumber, but there are zillions of different versions of these tests. Second, many practicioners have a prejudice against tests that are not copyrighted and commercialized by a publishing house; they feel that the logo of a publisher and the price to be paid are a sort of quality assurance - and this of course is a prejudice, but it affects the success of open source tests in professional practice.

https://openpsychometrics.org/

Here is an example of what you are suggesting in action.

If you wanted to develop an open-source test, you would need to build an initial framework and find collaborators willing to sacrifice time to develop it. Part of the high costs of test development goes into the salaries and operating costs to run the research required to develop and test the Validity and Reliability of the test.

Claro que sí, pueden ser con parámetros de criterio a las respuestas abiertas.

Are you thinking of specific tests? Some are freely available, including RAND SF-36, the PROMIS tools and others. It is certainly possible and has been done.

It is definitely possible. I'd recommend this paper for you to have a look.

Article An Adaptable, Open-Access Test Battery to Study the Fraction...