To answer the
part of you question. The answer is Yes, but it does not work the way you want.
You can take the URL of a timesheet and make that to a Favorite link.
Say you managed to sweet talk you Clarity admin to open a time period in the distant future and you saved you timesheet for that period with your 6 common tasks. That is fine.
You can open the the timesheet from the link.
There is a dropdown for the timeperiod and as soon as you choose another timeperiod the timesheet will switch to the time sheet of that period and you no longer have your 6 saved tasks on the sheet.