They used to be, however, in 1995 or so, they merged with "state police". They handle incidents occuring on state property, in addition to all the highway stuff. So, if something was happening at say, the DMV (RMV), they respond - be it a licensing matter or a domestic in the parking lot.
Since 1929, the CHP has had Statewide jurisdiction. If there is some doubt, remember the first word in the name of the agency - "CALIFORNIA" The unofficial motto is "All roads, all codes!"
To address take-home vehicle: All motorcycle officers take their bikes home; Captains and above have a take home unit; most investigators have a take home; Some special duty personnel have take homes assigned; some protective service personnel, who are on call, have a take home. The majority of the CHP officers live within a short commute of their assigned office.
The reference of the merger with the state police in the mid 90's was of benefit, mostly, to those State Police officers and personnel who received ample pay and benefits increases. The merger simplified the protective services detail and did away with duplicity of services. It did not change, alter, modify or increase the authority of CHP Officers. The one benefit to CHP officers is the previous name of "Traffic Officer" was changed simply to "Officer". That title was based on the majority of active personnel at the time who voted on whether they wanted the title to be Officer, Trooper, Traffic Officer, Agent, etc. Overwhelmingly, they opted for Officer.
At its creation in 1929, officers were known as State Traffic Officers. That changed to Traffic Officer in the 1970's and is now Officer.
An Officer's initial assignment out of the academy can be anywhere in the state. Most initial assignments are for a period of one year. After that an officer may request assignment to the area of their choice and assignments are based on departmental seniority. Pay grades are scheduled for every year for five years after successfully completing probation. There is a pay schedule on the site I provided which also lists other pay benefits - education, seniority, special assignments or skills (pilots, flight officers, motorcycles, investigators, etc.). Other benefits are nominal and well described..