Overview.
There are a number of settings that let you change the appearance of the calendar's tabs.
This will be available as a setting via an in-app-update: coming soon.
Changing the name of a tab is done the same way you alter any interface text: by editing DayBack's translation settings. This is really simple and you can learn more about it here: translation.
Yes. Log in as an administrator and you'll find an option for "Default View" under Admin Settings & Defaults / Views
In Settings / Admin Settings & Defaults / Views you'll see a setting where you can assign a new URL to the home tab: perhaps this is the url for your project management site or another site you use frequently. Leave the setting blank to remove the home button. You change the name from "Home" to something else by editing DayBack's translation settings.
This is also in Settings / Admin Settings & Defaults / Views where you'll see a place to pick the "Default Sidebar Tab".
Yes: this all done by editing DayBack's theme which is described in CSS. Tips and a video showing how we edit the CSS can be found here: CSS.