PFA = Popular Flying Association. All aircraft in the UK have to either have a Certificate of Airworthiness (CofA) issued by the CAA, or a Permit issued by the PFA. PFA aircraft usually include home-builts, classic aircraft, restoration projects, etc. They are cheaper for two reasons: modern homebuilts use technology which will take many years before all the legalities allow it to be used in CofA aircraft, and also the owners are allowed to do the maintenance themselves before having it signed off by an inspector. The
PFA's website has more information.
Whether you can get insured on a group aircraft will depend on the group. My group would insist on you having instruction with a PFA coach until he signs you off. If the insurance company increased our premium because of you, we'd expect you to pay the increase. The same rules apply to anyone who wanted to join our group. Other groups may have different rules - the only way you can find out is to ask!
Can't help you with the sponsorship question, but there's a new Sponsorship+Jobs forum where you should be able to get some advice on this one!
FFF
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[Edit beacuse I forgot to mention that many PFA aircraft are allowed to use MoGas - car petrol - which is cheaper than AvGas. And I'm sure there are other reasons why they're cheaper which I've forgotten, too.]