I am still not clear and hope you can help: does EVERYONE, who applies for the competitions get invited to go through the testing, or is there a pre-selection even there already?
At which stage EPSO conducts an eligibility check depends on the specific competition as well as what is stated in the notice of competition.
Short reply: Yes, usually everyone who applies for a competition gets to sit at least the first part(s) of the competition.
More lengthy reply: If we consider this year’s SC-1 & SC-2 competition EPSO/AST-SC/10/20, the notice of competition under “How will I be selected?” explains the procedure – and mentions that only after (1) the first application, (2) the preliminary computer-based test and (3) the intermediate test, (4) an eligibility check will be conducted based on the information provided in the application of those candidates that have received the best marks to make it potentially to (5) the assessment phase.
At the assessment phase, candidates will also have to bring proof of the respective documents necessary to meet the eligibility requirements – and the eligibility will be checked again for those candidates, but this time not merely based on the applicants' claims in their application but based on their documents.
At this SC-1 & SC-2 competition as well as many other competitions that have some degree of eligibility requirements (actually all have, even the generalist competition), everybody applying and validating their application (and having the languages of the respective competition) will get a chance to sit the computer test. A pre-selection/check of the candidates meeting the eligibility requirements takes place only at a later stage.
Applicants are however requested to provide accurate and truthful information already at the very beginning of their application stage. Of course sometimes, they arguably may not be in the position to determine whether for instance their work experience may be deemed relevant and thus "worthy" eligible to apply for a given competition. So, logically they can apply and it will be up to EPSO to determine the candidates’ eligibility.