Once you complete and submit the online form you should be given the option to download a document checklist. This checklist, along with application summary, referee forms, etc. will be available for you to download for at least a couple of months after submission using the same application login credentials.
The checklist itself is somewhat generic depending on the route that one takes throughout the application process, so it will not always reflect your exact circumstances (refer to Guide AN for specifics).
You do not need to send a BRP if you have never been issued with one. If you have an ILR endorsement in your passport, the passport itself should be sufficient.
Wrt to time, you have 10 working days from the date you submit your online application to post the document checklist with the supporting evidence to the UKVI address listed on the document checklist.
If you have chosen to apply with an appointment at UKVCAS, you will get to keep all of your documents as they will be scanned and verified for you by UKVCAS at your appointment.
If you have applied through the "existing" system,
all of your supporting documents will usually be kept until your application has been reviewed which can take up to 6 months. You will get your documents back along with your decision letter. That said, you can request to have your documents returned to you if you need to travel urgently. This can be done online
here and should take about 10 working days to process.
Edit: @CR001 beat me to it