No. Neither employment pass holders nor their employers (on behalf of EP holders) are required to make CPF or any other statutory fund contributions. CPF contributions are required only for Singapore citizens and permanent residents. CPF was established in 1955 and is basically a savings scheme to provide protection for locals (Singapore citizens and permanent residents) in their old age.
When your Singapore work visa or permanent residence application is approved, authorities may require you to undergo a medical test. You will receive an approval letter that will indicate whether or not you are required to go through the medical test. Typically, the medical test includes a physical examination and some blood tests (to rule out serious diseases) and would usually require you to undergo an HIV Test and a Chest X-Ray. The physician would need to sign a form that will be attached to the approval letter.
For getting the medical tests done, we encourage you to come to Singapore and go to any clinic or hospital where they can give you the report on the same day. Wherever you go for your medical examination, if time is of the essence to you, make sure you confirm when the report will be ready before you proceed with the exam. Also, make sure to check the approval letter on the latest government policies regarding the medical test.
If you would like to get your medical test done in your home country, you can do so through a reputable clinic/hospital and bring the report with you when you arrive in Singapore. Please be mindful that the medical report must be in English.