Should I send unsolicited applications for postdoc positions?
Absolutely you should, many positions aren't advertised and it could be your skills are just what the other group needed. Make sure you go for a quality over quantity approach, and really tailor each application to the group you are applying to. A well thought out cover letter (checked for spelling and grammar) and a well formatted and targeted CV puts you in a favourable position. A good approach is to send an application by post as well as by email; you will really stand out doing this. Talk about what you what you like in the group, your current situation, and how you think you would contribute to the group's research. In terms of funding, if you have some already great, if not, research what you would be eligible for and open the possibility of the pair of you developing an application together.