This is not a fact, but a common misconception, that the US gov and rightists actively promotes. To be far, it's common even on the left. The leaders were mostly students of proletarian backgrounds.
No, you have this question as wrong as you had the unproductive labor question. The BPP was founded by Bobby Seale and Huey Newton, along with a handful of others who are lesser known to the general public. Newton & Seale had gone to college, so that was part of their background. When they founded the organization, they were not in college. Since they were inspired by Fanon's writings on the anti-colonial revolutions and how central roles could be played in those by oppressed lumpen elements, the organization did orient themselves to that category of person.

This doesn't necessarily refer to "criminals" though it is true that Seal and Newton did break the law, as if that is necessarily a bad thing anyway. It refers to people are marginally employed, taking temp jobs or having no job at all over an extended time in a person's life so that it begins to define the logic of their self-reproduction. At the the time the BPP was founded, most of what became the leadership were in activism full-time (often not on a paid basis), though Seale had just found work in an anti-poverty program, a formative experience for how his politics were developing. A lot of this is discussed in the recent (and excellent) book Black Against Empire, and it shouldn't be surprising as one of the targets the BPP had set its sights on was incredibly high unemployment afflicted urban youths.

A lot of its members were teenagers with working-class parents. They were mostly not career criminals or in organized crime, nor were most hired muscle or agents of the states or bourgeoisie, except the snitches. Nor does some of the alleged "left" adventurism make someone a lumpen. Though they did seem to have underestimated the contradictions between the proletariat and the actual lumpenproletariat.
I never said that they were involved in organized crime, nor did I suggest that lumpen as a term was restricted to criminals.

And committing crimes, homelessness, poverty or being unemployed does not make one a lumpenproletarian, otherwise there would barely be a proletariat, if at all. As Engels wrote in The Condition of the Working-class in England
No, committing crimes doesn't make a person lumpen. Otherwise Wall Street executives would fit into that category. Being unemployed for prolong stretches, so that you begin to eek out a life trajectory outside of the formal economy, does make you lumpen. I'm not sure where you see a definition in Engels' early work there would contradict this fact, but I'd love to see a direct quote that spells this out (none of the quotes you provided do this), hopefully this one a little more accurate than your earlier "quoting" of Marx. Right now your rationale seems to be: "BPP = Good" + "Lumpen = Bad" therefore "BPP != Lumpen." This is what I meant in my previous post when I was talking about cramming and facts to fit moral categories, rather than basing morality off facts.