Yes you can apply small caps to the rest of the first line.
Almost the same way you set-up the Drop Cap, except you select the Font tab, then select the number of characters you would like changed to small caps. Also check the 'whole word' box. You can select the character style in a box on the same set-up screen.
EDITED: I failed to mention you need to set a new style by selecting Style Format NEW, Give this a name like 'First Line Caps.' While in the Organizer set the Next Style and Linked With. Back to this in a second.
Next go to Font Effects and turn on Capitals.
Then go to Indents & Spacing - First Line and set the desired amount of space after the drop cap. I'm not sure if checking the Automatic works for all width drop caps.
Here is the caveat. When you reach the end of the line with caps, you must use a paragraph return to invoke the NEXT STYLE which you will set without using First Line Indent, so the text does not get shoved in on the second line of the first paragraph. In essence, you are creating two separate paragraphs, but linking them together as one.
So you need your Drop Cap Style; First Line Caps Style; Rest of First Paragraph Style; and this last style must be linked to the Paragraph Style you are using for the rest of the paragraphs on the page.
I just tested this before replying and it worked just fine. I used a three line Drop Cap, with the rest of the first line all in Caps, and the second line of the first paragraph, forced with a paragraph return to use the Rest of First Paragraph Style. The next return brought me back to First Line Indent for the next paragraph in normal lower case text.
NOTE: This will not convert to e-books, but does save as a PDF for POD print books.
End of Edit.
If you plan on publishing as an e-book, because of the flowable text, it may cause parsing problems in the different readers out there. If you are going print, it converts OK to a PDF for POD printers like CreateSpace or Ingram.