Projects which I expect to list in the application were not done strictly as per PMP guidelines. Might not be able to get documents to show all the stages distinctly. Can I still apply?

Rest assured that PMI does not require the experiences of Aspirants to be based on the PMBOK® Guide framework (although if you work in an organisation practising the PMBOK® Guide project structure, you will find it easier to be well prepared for the PMP Exam).
 
The PMBOK® Guide describes the lifecycle of projects, major project processes and project groups according to PMI’s definition. One should be able to fit their experiences into the PMBOK® Guide framework. You do not need to have the processes/stages/groups named identical to the PMBOK® Guide on your project documents for PMI to count those experiences. Likewise, PMI does not require you to have the job title of “project manager” in order to recognize your project management experience. What PMI looks for is your actual experience in managing projects from start to finish — and you will inevitably have to deal with the project processes even without your own knowing.
 
In case of audit, you would just need to ask your (then) supervisor to sign the verification forms to validate that your stated experiences are valid. You would not normally be asked to provide the project documents, plan, etc. as these would be treated as confidential for most companies.
 
My recommendation is: just skim through the PMBOK® Guide and try to match your working experience with the PMBOK® Guide framework. Remember to make use of the terms in the PMBOK® Guide to describe your experience. In this way, PMI will recongize your experience more readily.
 
Wish you PMP success!
 
(further reading: https://edward-designer.com/web/online-pmp-application-form/ )