We see that today an Islamic ruling is derived using a set of guidelines and scientific process. Each of the four schools may have different rules to derive a ruling , so can this intricate process be a Bidah motivated by the hadith of Hair splitting in religious issues? Are there any sects or schools of thought which consider fiqh to be Bidah?
The Ahl-ul-Hadith movement of the 19th century, in the sub-continent, would qualify as a school of thought that comes very close to rejecting Fiqh. (Ahl-ul-Hadith not to be confused with primitive precursors to organized schools of Fiqh). Ahl-ul-Hadith do not ascribe to any school of Fiqh themselves and consider the association itself to be a Bida'h. One of their main theses is the absolute rejection of Taqlid, with an associated importance on deriving the rulings directly from sources. They are often compared to the now extinct Zahiri school of Fiqh but unlike the Zahiris the modern day Ahl-ul-Hadith don't consider Ijma to be a source.
Another school of thought are the Quranists. They reject the Hadith and tacitly the Sunnah as a source of guidance besides the Quran. Unlike Ahl-ul-Hadith, though, they are not particularly concerned with the concept of Bida'h.
Quranists are often classified as Modernists, though there are other modernists who accept Sunnah and Hadtih as source of legislation, while rejecting the organized schools of thought e.g. http://www.al-mawrid.org/
Very unlikely; no system of law can do without principles to guide judgement; and in Islam these principles are called Fiqh.
Bi'dah, is pejorative innovation in the Islamic Sciences; it can't be applied to the notion of Fiqh in toto.
(This doesn't mean to say that in day to day life, the 'Hadith of hair-splitting' won't be applied by the slightly over-zealous).