@ Matt
You can do some of this…
In the upper right corner of the browser window you will see a Screen Options tab. Click that. This will allow you to hide the metaboxes you do not need to show.
You can rename Biographical Info with the Say What? plugin.
Install the Say What? plugin.
It’ll add a new Text Changes menu item. Click Add New, enter the string exactly as shown in Connections, enter connections [notice the lack of capitalization, that's important!] as the domain and then enter your new text.
The great thing about using this plugin is that it will be update safe and you can use it on any other plugin that has been properly internationalized.
For Original string you would use Biographical Info and for New string you would use Employment History
The last item on you list would be custom fields. Currently this is not supported by Connections. The work is completed to add this support in the next Connections update. You can see it in action here. Keep in mind that this new support would still require coding to add the new fields as there is not yet a point and click way to add them.
I hope that helps.
