1. Any Confederate attempt to recover MO was doomed from the start.
2. Yes. It's resources. Any halfway competant Union commander could do the job, and the Federal government could, any time it chose to, send overwhelming force into MO. In essence, Federal control of the MS River gave what Napoleon always valued: "interior lines". They could transfer troops from the Cis-MS to the Trans-MS and the Confederates couldn't respond.
3. This is the real question. Given the resources in the TMD, a Confederate raid into MO, meant as such, could divert federal troops from the Cis-MS (as it did) and could accomplish some recruiting (as it did).
A more competant leader could have swelled Confederate ranks and brought those new recruits more or less intact to AR. At this stage 5000 TMD recruits, some reluctant, some not, couldn't effect the war in the TMD, let alone the wider war. And it is notorious the TMD didn't have enough modern rifles for the troops it already had, so the question is how those recruits could have be armed.
Steve Newton, in his "Lost for the Cause", posits an alternative to Price's Raid--an expedition to expel the Unionists from Little Rock and the AR River valley. This move was more feasible, logistically, and the Confeds could bring their TMD infantry (flushed with success after the Red River campaign) to bear on Steele's forces. I think the logistics would have broken down here, also, since the same ground had been traversed by Steele in the Camden Expedition and was "eaten up". Plus the recovery of Little Rock, while a Confederate plus, wouldn't have been decisive in the larger scheme of the war. Still, it may have been a move worth trying.