AU Payroll - Type and search in the Employee filter
The current screen and filters for Xero timesheet is not user-friendly. In particular, firstly, it would be great if the Employee Name filter allowed you to type and search. Secondly, if the Employee Name filter could include multiple names that would be useful.
Lastly, not in the filters, but the tick boxes for each timesheet - if the tick box was bigger and next to the line, instead of in the timesheet line this would help. The way it is at the moment, it is too easy to accidentally go into a timesheet instead of ticking it to select for bulk approval.

Thanks for raising with us in Product Ideas here, Yana. It helps to have each idea separate so others know what they're adding their vote for and we can share the demand of each back with our product teams. I've slightly amended the title of your idea here to fit your first idea.
In terms of employee selections this is actually already possible - From the timesheets filter just click on the names on the employees you'd like to filter timesheets for.
We don't appear to have an existing idea for your 3rd point, but welcome to start a new idea so we can begin to track interest of this, too.