1. During the 1980s and 1990s, Jeremy Corbyn supported the IRA.
No. He didn't support them, he talked to them, in an effort to bolster peace. When John Major did the same thing, the peace agreement came about.
2. On national television, Jeremy Corbyn refused to back a shoot-to-kill policy if a Paris-style machine gun attack happened in London. He then changed his mind and backtracked a day later
www.theguardian.com/politics/2017/jun/05/theresa-may-attacks-corybn-shoot-to-kill-stance-as-bbc-clip-is-shared-online#img-3
3. Corbyn’s botched attempt at a publicity stunt on a ‘ram-packed’ train which was questioned by Virgin who released CCTV images showing the Labour leader appearing to walk past empty seats before he had filmed a video showing him sitting on the floor of a train carriage. Another image released by Virgin also showed Corbyn having later found a seat
And other people on Twitter vouched for him in saying that he did not have a seat and the photo was clipped from after the train emptied.
I've been on that train many, many times. It is always rammed until you get to York and then it empties.
4. Many advisors and colleagues have resigned under Corbyn or disowned him in the last ten months, citing incompetence and his unelectability
I agree with you on this one. This is my concern, not that he's incompetent, imo, their issue with him is that he'd dragged the party back from being Tory lite towards true Labour values, however, yes, the lack of support he has in his own party is worrying.
5. Every large-scale study into why Labour lost the 2015 general election came to the same conclusion, Labour was not trusted on the economy
That is people's perception, yes, but it doesn't necessarily mean it's true. Historically Labour borrow less and pay back more.
If you're gonna denigrate the man, at least do it on actual facts and not media soundbites.