1. Dear Di, the name given to the person sued for personal injury is a "Defendant" not a Respondent. Secondly, there can be more than one Defendant to a legal action for personal injury. Thirdly, you can sue both the owner of the dog and the management company, so long as you can show fault on their part. However, you cannot sue the owner of the house where the tenant resides as there is no fault on their part for the dog biting the child. Just because a tenant and a dog happen to live in a property you rent, this gives rise to no liability for what the dog does. I would advise you to see a solicitor here. Be aware that the Gardai may seek to prosecute the owner of the dog.
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3. You are confusing two issues here. Insurance and liability are two different issues. A person can be held liable irrespective of whether they have insurance or not. You are focusing too much on insurance. It is a question of fact whether fault can be attributed to the management company. I cannot give you an answer unless I had the full facts before me and had the management contract of the management company. You should speak to your solicitor (and barrister) about this issue. Finally, the criminal and civil cases are different strands. However, if a prosecution of the owner is successful, liability for civil liability under the Animals Act would follow. However, it would only apply to the owner. Finally, you are correct about the time limit. However, here, you should speak to a solicitor as a matter of urgency. Don't delay.