I believe the exam is linked to your account so you should be able to log in on the school's computer and take the exam even if you paid for the exam on your laptop.
Yes, just be sure that the tangix software is loaded on the computer you take the test with.
You first create an account (can be done from any computer / device that will let you connect to Solidworks.com.)
At that time or any time later you can log into your account and navigate to the certification center - again from any device. you can purchase exams at any time and they are deposited into your account as a credit.
At any time you want you can download the Tangix software and read the information.
There are many good suggestions and guidelines that the initial documentation will explain.
You can also take a practice exam (free) so you can see the methods used.
Read all the documentation and tips there and setup your templates and file locations before actually retrieving your exam.
The tests are all timed tests and if you stop or get interrupted or your computer crashes you will have to re-purchase the test -your credit is used up when you first start the actual test - so be ready.
You MUST have a reliable internet connection during the entire test. If the internet connection goes down the system will try to reconnect for 2 minutes. If it can't your test will terminate and it will create a file that you must send in. ( I don't know what happens next or it you are allowed to continue where you left off or anything else about that)
All this information and suggestions are in the documents that are on the web site and in the tester setup.
Best is if you can have dual monitors or you can use 2 computers or devices next to each other (you must be able to transfer files back and forth (only one (the one with the tester program) needs internet connection - the other (if using 2) would be running Solidworks. Be sure you have a time and location you won't be interrupted as once the time starts it can't be paused.
I suggest also doing some research on the tests and the testing - there are many videos and books available.
EDIT: Do NOT try to do the test on a tablet or something with a small screen and you should be using a mouse if using a laptop for easier movements.