The layout has nothing to do with CBSubs. The layout is 100% generated by CB which is 100% open source. At any rate the above can be accomplished using purely CSS or you can alter the registration layout at the below file.
components/com_comprofiler/plugin/templates/default/default.php
The fields rows are generated by the field themselves so you may not be able to edit the layout to the extent you're wanting without altering CBs core files, which I can't advise on doing.
Additionally, is there a quick way of making the registration fields NOT have the alternating background of grey and white?
Yes, with the below CSS.
#registrationTable .sectiontableentry1,
#registrationTable .sectiontableentry2 {
background-color: white;
}
You can change the color to whatever you like or try setting it to transparent. I recommend adding any custom CSS to the bottom of your Joomla template CSS file.