Of the ships you list, the viable ones for Level 3 security missions are the Praxis, the Drake and the Myrmidon. You need a Battlecruiser at least - a cruiser can run them, but it takes a fair bit of skill and picking the approach carefully.
Given you’ve a Vexor and you are a Gallente pilot then the Drake is probably a lot of cross training - it’s a shield fitted missile ship, and if you have been following a typical Gallente path you’ll have focused more on drones, hybrids and armour systems. So, if that’s the case it’s probably not going to be a big passive tanked Drake. Shame, because it hammers Level 3 missions.
The Praxis is a slow compromise of a ship. I’ve not flown it, but it handles all weapon systems and all modes of defence equally “well”. It’s not as good as a focused battleship - lower firepower and lower defence than others in its class, but is still a step up from a Battlecruiser. It may be a little frustrating because it’s the slowest of the ships you have listed.
I run Level 3 (and 4) missions in a Heavy Assault Cruiser - Bigger stuff now feels really slow! - but that’s a lot of skill training, some of which is Omega only, and a 300m ISK ship.
The Myrmidon I don’t know well, but depending on your skills it is probably the logical choice - ideally with T2 drones and a good active armour tank. You should only need medium and light drones - keep an eye on them as NPCs - especially Elite Mercenaries - do like to attack them.
Looking at the University fits they do make a good case for a passive shield fit for missions - but you may not have the skill set for that to be effective (if you did then the Drake is better!).
The advantage of a drone boat is you can change the weapon size easily: medium missiles (Heavy and Assault) aren’t brilliant against fast moving frigates (they will kill’em eventually), but you can always put out a flight of light drones to deal with the frigates before switching to medium drones to kill the bigger stuff.
For high-sec ratting the same ship as you use for Level 3 missions should suffice. Indeed it’ll probably be overkill. You may want to carry a flight of salvage drones and tractors if you intend to make some additional ISK from the wreckage you create.
As has been noted, Level 4 missions are now Omega only (people were farming them). But Level 3 missions are a good modest earner. Find a nice mission base in a 0.6 or 0.5 system, ideally not to close to lo-sec. The lower security areas have better rewards than mission running out of a 1.0 system.
As an aside: that was a well thought through question: What you wanted and your limitations. Thank-you.