According to the California Department of Motor Vehicles if you are a visitor in California over 18 and have a valid driver license from your home state or country, you may drive in this state without getting a California driver license as long as your home state license remains valid. So you could not even drive into California with your suspended license in Nebraska since it is not valid.
Once in California you can apply for a CA license but the DMV states that the license applicant shall submit to an examination appropriate to the type of motor vehicle or combination of vehicles he or she desires a license to drive, except that the department may waive the driving test part of the examination for any applicant who submits a license issued by another state, territory, or possession of the United States, the District of Columbia, or the Commonwealth of Puerto Rico if the department verifies through any acknowledged national driver record data source that there are no stops, holds, or other impediments to its issuance.
Your suspension will be listed on the National Driver Register (NDR) and the Problem Driver Pointer System (PDPS). In fact all states check the Problem Driver Pointer System and/or National Driver Register to see if a driver's license applicant is listed.
The PDPS helps search the NDR which is a central database of information on problem drivers (those whose license has been revoked, suspended, canceled or denied or have been convicted of serious traffic related offenses) provided by all 51 US jurisdictions.
Based on information received as a result of an NDR search, PDPS will "point" the inquiring jurisdiction to the State of Record, where an individual's driver status and history information is stored. The PDPS is intended to assist jurisdictions in the meeting the basic tenet that has been in the works of the US that "each driver, nationwide, have only one driver license and one record."
So it is our understanding that your suspension in Nebraska for unpaid tickets would prevent you from obtaining a California though you can find out for certain by contacting the CA DMV. You will likely need to find out how to get these tickets paid and your NE driver's license reinstated in able to apply for a CA driver's license.
The unpaid tickets may not affect your insurance at this point, that is until you pay the tickets and are convicted of the violations on the tickets, but your license suspension will probably affect your insurance rates.
Once you have a valid license you can obtain auto insurance quotes online with us here.