You will need to use the data in Alteryx workflows to find the answers to the practical questions - so yes, the files must be saved to your laptop.
For the practical questions, for most computers, if you click download, the data automatically are saved in your Downloads folder. The dataset is specific to each question. As you attempt one question at a time and save data, the files can be sorted in in download folder in chronological order.
Don't get disheartened. Actually taking the exam is part of the learning process - as you find out more 'known unknowns' and reduce 'unknown unknowns'
I have coached team members in Alteryx and some of them passed the core exams within 1-2 weeks - with no prior experience.
The common denominators for successfully passing the Core exam seems to include:
1) Read the Core exam guide, and make sure that you know every single tool listed there.
2) Completed all the interactive lessons that are marked as "CORE".
3) Do at least 20 weekly challenges. This is essential that you can apply (and hence solve practical application questions).
4) Some also find the recorded videos on Core Exam Prep (5 hours or so) helpful.
Good luck next try and happy learning.