View grid mode won't work
Your theme could be breaking the grid. Grid's are usually tables, so they generally work regardless of the theme - but if your theme is making any modifications to it the grid may break. I would suggest examining the HTML code that is being generated for the grid (in Chrome: right click the grid and select Inspect Element). Make sure that it's a table... and also examine the styles applied to it to see if any of them are breaking the normal grid display.
No - you can't tell Views to put certain fields in the grid and certain fields outside of it. I can't really imagine how that would work, or what you mean by that exactly, so it's hard to suggest an alternative. Perhaps two separate Views?