I think the sole idea for a share was the fact you could take the aircraft to Portugal for a week and no-one else in the group would mind...so make sure the airctraft is what you want for the trips you plan on doing.
Make sure the group already has a substantial fund - e.g. £10K +, that it is well run, and has a set of legal rules for which you sign up to, including rules on how you sell your share. A well run group will have all these adminstrative things sorted out. Check the items that are due to be replaced, etc, and when the next C of A is etc; obviously the thing to do here is to avoid being faced with a huge cost of a replacement engine the day after you signed up. I was negotiating a opt out clause for the first 6 months, so that I was immune from some disaster.
The more complex the aircraft the more fussy the current members are on who joins. This can be driven by insurance, but many will request a PPL and IMC. Any increase in the insurance because of you will see you footing the bill for that increase.
Sounds like you want to do some extensive flying, so have a nice comfy aircraft, with a good set of avionics. The aircraft I was looking at (a Commander 114) had a recent overhaul of the avionics, which cost about £30K.
As for mentioning that you want to hour build, be aware that may make some people run a mile, and refuse to accept you.
Also, you will probably become a director in the company that owns the plane, so make sure you're not a bankrupt!!