Yes, you may apply for change of status while on H-3 visa. If you are taking up a job, have your employer file Form I-129, Petition for Nonimmigrant Worker, at the USCIS Regional Service Center with jurisdiction over your case if you are trying to change to
H-1B or TN. Make sure your employer files for your spouse and children on Form I-539, Application to Extend or Change Nonimmigrant Status.
No, there are no travel restrictions on H-3 visa. You may travel as many number of times as required before the expiry of your H-3 status. The USCIS also does not impose any time limit on your stay abroad.