Assuming that you are with your EU-citizen husband more or less continually, you fall under the freedom of movement directive 2004/38/EC. This might not apply in the UK or Ireland, however, because of his having both nationalities. Since the seven cities you mention are all in different countries we can ignore that question.
if we are moving countries, would I need to register in every country for 3 months?
Under the directive, each EU country can require you to apply for a residence card if you stay for longer than three months (Article 7). If you stay for no more than three months in any one country, there should be no need to apply for the card (Article 6). In addition to the residence card, a country can "require [you] to report [your] presence within its territory within a reasonable and non-discriminatory period of time." This allows countries to require you, for example, to register your residence, and "non-discriminatory" means that the requirement should be the same as any such requirement that exists for citizens of that country.
Do you think that, after a period of time, I am going to get in trouble or get deported?
The directive states (implicitly) that you cannot be deported for failing to register or failing to register or failing to apply for a residence card; in each case the penalties are limited to "proportionate and non-discriminatory sanctions" (Article 5(5), Article 9(3)).
Can I work freely, as a contractor, without breaking any rule?
In theory, you should be able to work, but the administrative formalities required to document your right to work might be impractical (they would be, for example, in the UK). I do not know about seven countries you are contemplating.
In particular, it might be difficult to work before you have a residence card. But if you're working as a contractor, it is likely to be easier, especially if you're being paid by a company in a different country. For example, it's unlikely to be a problem if clients in the country where you're staying are paying your husband's company in the UK, which in turn is paying you.