Who owns the equipment is not relevant. What matters is that the photography complies with management policy and permissions.

There is no copyright in animals (although Patent law can apply to selective breeding and genetic modification, but you aren't doing that). The pet isn't going to sue ;-)

You almost certainly have a normal expectation of privacy in your back garden. So no, your neighbour had no legal right to take such a photograph without your permission.

It depends on the circumstances of the photography, the agreement (if any) made with the subject eg a model release, and the usage to which the photos will be put.

Check the park bye-laws. Most prohibit photography for any commercial purpose. However you are unlikely to get into any trouble, the penalty is usually about £5.

You certainly do : you own the copyright of your own work. Please see the FAQ at our sister site http://copyrightaction.com/faq for relevant information.

There are several issues here.

Usually a subject will have no ownership rights over a photograph. As the author of the work you are the copyright owner unless you have made a contract with them that assigns copyright to them.

CRB checking is designed for employers to vet prospective employees or contractors who may at some point have access to children in education and child care contexts.

In general minors have no greater or lesser rights than adults, and in general editorial photos taken in public places will require no consent.