Sort of true. They will do the trade, but only if there was already someone in the system willing to make the same trade. HFT firms don't hold positions at the end of the day.
AIUI, the difference is that, at the moment the HFTs place the order, they have every intention of filling them. Just because they might change their mind a fraction of a second later and modify their orders isn't particularly relevant.
Whereas spoofing is placing orders that you don't have any intention of filling. The only point of the orders is to move the market, rather than to actually make trades. And that's the thing that's illegal.
Well, the law doesn't really say anything beyond "manipulative devices" and "contrivances," but right now, judges have kinda ruled that intention matters. But we don't know much, because most of these cases get settled out of court, so we don't have a robust case law to base any of this off of. Basically, if you participate in a financial transaction in any way, be prepared to be accused of a crime.
Intention mostly matters, sometimes doesn't. Sometimes the left arm doesn't talk to the right arm and conflicting orders go out. Technically this could be construed as spoofing, but sometimes it's business unit A not talking to business unit B, because A isn't in the same location as B, or there's a Chinese Firewall between A & B. It might not be intentional spoofing, but it can happen, anyway.
No, they are allowed because HFT firms have the intention to (and in fact, will gladly) trade.