I’m a Jeremy Corbyn supporter and I have some ideas, numbered respectively to your questions —
1) As you mention, during the Sadiq Khan campaign Londoners ultimately ignored the lies and attacks from the Goldsmith camp and backed Khan. Can Corbyn and co. analyse how his campaign overcame the onslaught and apply a similar strategy?
2) Labour should reference what the Attlee government achieved more prominently, to show what is possible with a left-wing government, and promise similarly ground-breaking progress.
3) For one, nationalisation of the railways and any other industries needs heavily promoting and explaining in more detail for all to understand how it will benefit them and their communities. Price cuts? More jobs? Increased and easier mobility? On what scale?
4) Can Labour take a leaf out of the Conservatives’ book and launch a targeted advertising campaign, in mainstream media as well as on social media. As asked by you — how can Labour win over Conservative voters or older demographics? If we find some answers, how about creating some policy based on these results and taking out some ads in The Times, The Telegraph, The Daily Mail etc to share the message? Probably The Guardian too for that matter.
I’ll have a think on the remaining 4.