I'm prior-service Army, having served four years and two combat tours. I can't speak as to certain MOS's or their prerequisites--I was a groundpounder, with a few months light duty in a shop until I recovered from an injury. So there's my background and credentials for this subject.
First a brief outline of the enlisted rank structure, from lowest (e-1, "private") to highest (e-10, "Sergeant-Major of the Army"):
e-1: Private, first grade 1 (PV1)
e-2: Private, second grade (PV2)
e-3: Private First-Class (PFC)
e-4: Specialist (SPC)/Corporal (CPL)
e-5: Sergeant (SGT)
e-6: Staff Sergeant (SSG)
e-7: Sergeant First-Class (SFC)
e-8: Master Sergeant (MSG)/First Sergeant (1SG)
e-9: Sergeant-Major (SGM)/Command Sergeant-Major (CSM)
e-10: Sergeant-Major of the Army (SMA)
Depending on one's schooling, a person could expect to leave Boot with any ranking between e-1 and e-4. Having a BA would net you a specialist-rank, but in many jobs would have you labelled a 'college boy' who didn't earn your stripes. Even in mostly-clerical positions you could expect to catch a certain amount of flak if you jump in with your 'sham shield' straight from Basic.
Now, in order to progress past the rank of specialist you go before a Board, made up of the First Sergeants of each company/battery in your battalion, the CSM of that battalion, and possibly the ranking officer himself (generally a 'full-bird' Colonel: which is just shy of Brigadier-General, or a 'One-Star'). There are certain things you're expected off-hand to know: such as the soldier's creed, the Army Song, the 5 paragraphs of an Op Order, the 8 function steps of the m4 carbine... the list kinda goes on and on. But if you answer the questions to their satisfaction--and beat out the few-dozen or so competing e-4's--then you get a fancy new rank and more responsibilities! They can expect to serve in their new function for a few years, until the time comes to rank up once more (mandatory 6-year period between the e-5 board and e-6). This is assuming an unblemished record: otherwise you could very well get bumped right back to the start. On my first deployment we had a SSG whom no one particularly cared for. A year down the road one of my constituents ran into them within their next Unit, and he had gone from e-6 to e-1 for harassment issues. We all had a good laugh at that! But anyway, the point I'm trying to convey is it's not all about time in service and accomplishment: there's also time-in-rank, good conduct, and a certain level of general/technical know-how involved.
As Valerian said, being a sergeant has a certain amount of desk-piloting to it. But unless your MOS is mostly office-oriented you can expect a lion's share of boot-time. Within the infantry an e-5 was the rank most enlisted got their first real command at: they could expect to control a 'fire team', consisting of about four subordinate soldiers, at that rank. They would still have paperwork to accomplish, but for the most part it was overseeing and directing younger soldiers. And again, beyond that lies the rank of SSG. This rank can expect to have between two and four sergeants under them, with all the other subordinates that entails.
As to your first question? A soldier who entered Boot with a BA, and subsequently got two tours under their belt, could reasonably expect to have the rank of sergeant under their belt. This would entail having an e-4 ranking straight from Basic and AIT (Advanced Individual Training: which pertains specifically to the job you're contracted to do), and not coasting under the radar for the two tours of duty. Theoretically, one could ride out a total of nine years and never see the rank of e-5: but most soldiers start to crave that bump in pay, especially after they realize that your junior enlisted-status doesn't necessarily protect you from doing an NCO's job. The MOS's you listed are themselves largely interchangeable: unless by 'paramedic' you mean 'combat medic'. Being a medic and a 'Doc' are two entirely different things. But regardless, everyone can expect trigger-time once they get downrange. To coin one of my most-revered squad leaders: 'Hajji don't give a [EXPLETIVE] if you work in a shop or not!'