Yes. It's basic democratic principles. And, yes, they did run basic checks on people in previous elections.
Anyone who joins whilst being a member of another party is a dishonest turd, and should be removed from membership by both.
This is a vote for the only potential left of center challenger for prime minister there is likely to be, so ultimately it's a vote for who I'd prefer to see as Prime Minister. Now the SNP have the whole of Scotland, the chances are this will be a coalition or minority government with confidence and supply by other left of centre parties, but either way in any successful challenge to the tories, the leader of the Labour party will become the prime minister heading up either a majority labour government, a coalition or minority labour government.
We don't get to vote directly for prime minister in this country, so this is the only real chance any of us have to directly influence that selection of the person I'd want to end up as prime minister.
The Green Party would probably offer confidence and supply arrangements to any minority labour government, definitely with Corbyn as leader, whereas it wouldn't for a Tory government. So I'd see us more as potential allies than opponents, and with a less partisan none neoliberal leader that alliance could be put on a more formal basis.
Not that the Green Party would amount to much with Corbyn as Labour leader as we'd lose a hell of a lot of votes and members to a centre left labour party, so I doubt we'd have more than the 1 MP we've got now.
The best I could hope for with the Green party at the last election was that it would pull Labour to the left, and potentially help counter the prevailing austerity narative - which I think happened to some extent. +I suspect that the surge in support for Corbyn's position now has been helped a lot by the inclusion of real anti-austerity voices in the debates (SNP being the strongest, but the Green Party activists were making that case at a more local level across most of England).
ps I was working on the assumption that being a registered supporter was different to being a party member, if not then why set that category up?