Similarly, there's no legal requirement for anyone contracting for a publisher, on a freelance basis, to have any form of insurance. However, the publisher might have insurance against, say, claims for libel. Such insurance might only apply to employees of the company. It would not be unreasonable for such a publisher to require than any contractor has their own insurance (indemnifying not only the contractor but the publisher against any possible claims). It wouldn't be a legal requirement but neither would it be an unreasonable requirement from the publisher. (Although,as an occasional freelance writer myself, I've never heard of anything like it).
Any contract of employment is entirely separate from any other contract which may have previously existed between the two parties. You're not obliged to accept any particular rate of pay and the publisher is not obliged to offer any particular rate (other than a minimum equal to the national minimum wage). You can demand what you like (at the risk of being refused by the publisher) and the publisher can offer what they like (at the risk of being turned down by you). It's all down to your negotiating skills.
Chris