I don't know, is it possible or not, to know if the browser app is currently open.
But, yes, you can get the page that is viewed. That is all the pages searched, browsed or bookmarked. And then, using that url, you can get the page.
Luckily I have done an application to parse html of page in browser.
The code to get the url of last page viewed in browser is:
Cursor c1 = Browser.getAllVisitedUrls(getContentResolver());
Cursor c2 = Browser.getAllBookmarks(getContentResolver());
c1.moveToFirst();
c1.moveToPosition((c1.getCount() - c2.getCount()) - 1);
String url = c1.getString(0);
c1.close();
c2.close();
Url is what you needed.