Good news! Inbox on mobile already exists. You can find it by selecting settings (the cog wheel in the top-right) and then "Inbox" from the list. It's here on iOS and Android.
If you meant to ask, however, whether it would be moving that to the Activity tab instead, then we can confirm there are no plans to do so at this time.
As for the timestamps, it might be that you are missing the timestamp because of the dashboard setting:
Toggling that on should show the timestamp on asks as well as the dashboard. On mobile, you can find this toggle by selecting settings (cog wheel in the top-right), and then "Account settings", and then "Dashboard preferences" from that list. You will find the timestamp toggle in the list that follows.
It is much the same on desktop: find the settings tab from the left-hand scroll menu and then "Dashboard" from the menu on the right, where you will see the timestamp toggle.
We hope this helps. Keep the questions coming, folks!