The Picture of Dorian Gray/Headscratchers

The painting was Dorian's Soul Jar, and he probably knew it (or not), so why in heck did he freak out when he learned that he was on James Vane's hit list?

Given Our Souls Are Different, Dorian probably was unkillable by any 'normal' means, like getting shot to the head.

Note that Dorian still suffered from the withdrawal of his opium addiction and doesn't seem immune to diseases, plus feels the strain of his age. Thus it seems the painting only hid things; it could not erase them entirely -- so he could easily either have died from old age or been killed.
The limitations of the portrait's ability were somewhat expanded in his (non-canonical) appearance in The League Of Extraordinary Gentleman, where the magic of the portrait literally protects Dorian from physical injury.
It may be that Dorian was more afraid of being hunted by a ghost from his past that knows what he did than any physical harm that James Vane might visit on him.
He asked that the painting not make him look old or wicked. I don't remember him throwing in a "Hey, make me bulletproof too" line.
Even if the painting could, conceivably, protect him from death, Dorian doesn't know that it would. He never really tested its limitations. Why risk it?
Was he immortal? Would he had lived for ever, or was he really aging?

Why was Basil Hallward ever friends with Lord Henry in the first place?

We see in the very first scene they're in that Basil knows the guy is bad news, and that he disapproves of Lord Henry's warped hedonistic philosophies and despises the poisoning influence he has on people, so why would Basil ever willingly associate with him?

As alluded to in the article, both characters have facets of Wilde's personality (Basil is the friendly dedicated artist and Henry is an exaggerated version of the perception of Wilde as a cynical wit). So, it kind of makes sense that the two would hang out, since they are in a sense, the "same person". Outside of that, they have some shared artistic preferences and are likely both gay, so they probably move in the same social circles- although that doesn't really explain why they are friends and not just acquaintances.
Reread some of the early dialogue between Basil and Harry - Basil is pretty convinced that Harry is exaggerating his vices, "never [says] a moral thing and never [does] a wrong one," etc. Lord Henry seems like a normal enough (though very cynical and rather tasteless in terms of humour) guy to Basil. Basil may be overly optimistic about that, or he may not - the jury's still out on how "bad news" Harry really is. Aside from lampooning Victorian social mores and encouraging Dorian's vanity/paranoia about aging, we don't see Lord Henry doing anything particularly evil, and he seems to have no idea of what Dorian is really up to by the end of the novel. Lastly, Basil's concern about the role Lord Henry starts to play in Dorian's life could easily be put down to jealousy, rather than knowing that Lord Henry is *evil* per se.
I've always seen it as Basil knowing Henry well enough to know while Henry talks a lot, he's just trying to be shocking. Unfortunately, nobody told Dorian it was just an act. While Dorian later became very sneaky, he still lacked the common sense Henry and Basil had, he might have actually believed Henry really did do the horrible deeds he spoke of, hence blaming Henry for his own damnation.

Why is Lord Henry punished at all? He's the only major character alive by the end of the book

I'm slightly confused by the question, but I'll try to answer it as I understand it! He is indeed punished (since his two arguably closest friends are now dead) and he is punished because he is the one who really started this chain of events in the first place. He knew of his influence over Dorian and corrupted him, leading to his demise. He was the one who told Dorian that the only things worthwhile in this life are youth and beauty. Had he not said that, it's likely Dorian would have never made his wish and everything would have been hunky-dory. But he did and his friends died, along with at least three others known and one other alluded to. If he ever realized his part in this tragedy, he'd have to live with that for the rest of his days. If not...well, he's divorced, he's getting old, and has no Dorian or Basil to balance him out. Sounds kind of sucky to me.
Sorry I meant why Lord Henry wasn't punished at all. But you answered my question thanks.

Did Dorian love Henry the way Basil loved him, or was he just tempted by the lifestyle he showed him?

Though, the script did write that Henry was too clever and cynical to be really fond of.