There is no difference between the two in Canada. Public Liability Insurance is a term that is more popular overseas, especially in the UK and US. In Canada, the same coverage is provided by Commercial General Liability, often simply referred to as General Liability Insurance, CGL, or even just Liability Insurance.
No, Public Liability Insurance only provides coverage for accidental injuries and property damage to third-parties. Errors and omissions in service are covered under a separate policy called Professional Liability Insurance, also often referred to as Errors and Omissions Insurance. If a client were to be injured due to a service you provided, you would require Professional Liability Insurance to cover you in such a scenario.
No, Public Liability Insurance is designed to protect you in claims from third-party injuries or property damage. Your own property is not covered under this policy. If you own expensive property and equipment and require coverage, discuss with your broker to get Contents Coverage or Equipment Breakdown Coverage.