There's a number of ways to go about the design of such a class, though the mention of health management of constantly-decaying minions makes it a lot more complicated. One of the biggest things would be having the largest sources of redundancy, with regards to healing and AoE buffs, be present on items, and have other Heroes also get stuff where this is important. Maybe throw in Beastmasters and some sort of tinker and have mercenaries/followers of some description, to make such things useful regardless of Hero.
As for ordering the minions, we're talking about Warcraft 3. Unless you're wanting dozens, plural, of each of multiple kinds of minion (at which point the pathing is going to be more of a barrier than selection), then you can just be using the RTS gameplay interface to select and attack whatever you want, instead of locking to an ability. Which also allows for the minions to have some activatable abilities of their own without needing the Hero's interface bloated by other needs.
With regards to the actual particular "style" in question, a key would be multi-purpose abilities, filling multiple "roles" with one slot, such as having life-on-hit be attached to a Parasite-type ability, summoning the "chaff" units and applying a DoT alongside the heal on those who attack the target. This allows for having a lot more options in how you level up abilities and makes quick function easy, so you're not dealing with having to get a point in every ability just to be working smoothly.