Hi Iakov,
It is possible, but some of the tracking would require a trusted function, which is difficult to deploy and maintain.
Tracking when the form is opened is easiest. You can use the doc:Ready event, because this only fires once. There is kind of an example here https://acrobat.com/#d=06u-J6N34PQdaD25ZCPrSA, where objects are tracked, it may give some ideas for the docReady event.
There is a docClose event, but I think this fires after the form is closed, so you probably won't be able to write this data (time) to the form itself.
Accessing the user name will require the trusted function, there is a thread here dealign with that: Keep user from printing and seeing certain pages
Hope that helps,
Niall