I have heard that the best external networking is done at the client site during the audit. Is it possible/ common to secure a job in industry through your contacts this way?
Yes, a huge advantage of audit (and in the Big4 specifically) are the external networking opportunities, depending on which client you work on. You spend a large amount of time with these people and if you have a good relationship, you should ensure you keep in touch (absolutely add them on Linkedin after the audit year end has finished).
In terms of your second question - something to keep in mind. There is pretty much always an independence issue if you have worked on an audit of a company and then you want to work for them. I believe there is roughly a 12 month window in which you have to wait to work directly for a client.
I hope this helps!