You issue orders to each character for what you want them to do that round. Then you hit Start Fighting (or Start Combat, depending) and all of your characters and the enemies do their stuff for that round. You get to watch what happens but can't otherwise intervene until the end of the round. So it's the entire group of enemies and characters for one turn, not one character per turn.
For Wiz 8 there's a weird pseudo-realtime mode, but it's finicky at best. I generally stick to turn-based most of the time, and only switch to realtime to make sure my spells land in the right spot or change melee targets, and then immediately switch back.
In 6+7, when you gain levels you get random stat increases and HP and stamina and mana, and then you receive a bunch of skill points to allocate wherever you see fit. Sometimes you can get a +1 to most or all of your stats, and sometimes you get nothing or +1. Attributes are capped at 18 (but can be pushed to 20 with magic items) and skills are capped at 100. Most skills increase as you use them, though, especially weapon skills, so you still feel dynamic even mid-level.
In 8, you get a fixed (small) number of attribute points and a fixed (moderate) number of skill points... but you can only spend at most 3 points on any one thing per levelup. Your HP/stamina/mana is derived from your stats and not random. Everything goes up to 100, but can go to 125 with magic or items or specialization. Your skills can still go up as you use them, but they go up a LOT slower, and at levelup you can't put points into anything past 75, so your increases from 75-100 only happen through usage, which can take a while.
1 Each character can perform up to 3 attacks per round (depends on the level of the character)
This depends on which game you're playing. In 6+7, you can get at most two mainhand attacks and two offhand attacks per round, with a max of 4 strikes per attack. Non-combat-oriented classes may not get that many, and it's dependent (more or less) on which class you were when you hit level 20. If you were a Mage until 20 and then switch to Samurai, you won't get the full number of attacks. If you're a Samurai to 20 and then switch to Mage, you WILL get the full number. Two-handed weapons cause you to miss out on your offhand attacks entirely (so they're limited to a max 8 hits per round) but some two-handed weapons like Excaliber, Zatoichi Bo or Elven Bow+Mystic Arrows are totally worth it.
In 8, you can get a max of 3 mainhand attacks and 2 offhand attacks, of up to 3 strikes per attack. This is dependent on class, but has more to do with which class has the most class levels. The same rule for two-handed weapons applies, and again there are a lot of good two-handed swords and polearms out there.
In both cases, how many strikes you get is dependent on your class, your level, and how high your Dexterity and Speed are. In 6+7, Dex gives you more swings per attack and Speed gives you more attacks per round. In 8, it's reversed.