1. Radius Patterns follow Grid Lines

The ring version of Wall of Fire says this about its area of effect:

a ring of fire with a radius of up to 5 ft. per two levels

If you turn to the back of the DMG (p. 307) or look at this answer, you can see some patterns for radius spread effects. All of them spread along square edges. They're a ring of squares, rather than a circle. In this pattern, a square can't be half in or half out, because it only travels along the lines.

(This is actually also true for line effects, as PHB p. 176 shows. If a line goes through any part of a square, the entire square is affected. For something like Wall of Fire, an entire square has to be in one state or the other to follow with this, but if you place it along the grid lines it's fairly easy to do that.)

2. Yes

As you noted, Blade Barrier doesn't give a thickness or anything to imply that it's more than a curtain of blades (in fact, it says its a curtain of blades). You can treat it the same way as Wall of Fire: draw it on the grid lines. (The rules for area of spells mention that you should draw it starting from a grid intersection, so you can follow the grid lines.)

Regardless of the shape of the area, you select the point where the spell originates, but otherwise you don’t control which creatures or objects the spell affects. The point of origin of a spell is always a grid intersection. When determining whether a given creature is within the area of a spell, count out the distance from the point of origin in squares just as you do when moving a character or when determining the range for a ranged attack. The only difference is that instead of counting from the center of one square to the center of the next, you count from intersection to intersection.

This also makes the cover part of it easy to handle: anything on one side of that grid line attacking the other side has to deal with the cover portion of the spell.

3. Creature Size Matters

If you're placing it on grid lines (as you should), then yes, you can't actually place it on top of a medium creature. Large (or bigger) creatures take up multiple squares and you could have the barrier cast such that it appears in the middle of one. That makes this part of the spell make sense:

If you evoke the barrier so that it appears where creatures are, each creature takes damage as if passing through the wall. Each such creature can avoid the wall (ending up on the side of its choice) and thus take no damage by making a successful Reflex save.

Medium or smaller creatures wouldn't be damaged, as the barrier is not "where the creatures are".

4. Because the spell says so

This point has no rules-as-written answer, except because the spell says so. :)

I'd speculate that the game designers did that with the reasoning that as the spell comes into existence, someone can react and get out of the way before it can fully attack. Whereas once it's already there, someone has to go through it. But I don't know of any rules, anywhere, that explain why they did it this way.

5. Yes, you can attack

Blade Barrier says this:

Any creature passing through the wall takes 1d6 points of damage per caster level (maximum 15d6), with a Reflex save for half damage.

By a strict reading, passing through the wall is movement. By that reading: yes, you can attack through it without taking damage. I'm not aware of any rules clarification on that point.

Anybody attacking through it does have to deal with cover:

A blade barrier provides cover (+4 bonus to AC, +2 bonus on Reflex saves) against attacks made through it.

The wall is opaque but doesn't otherwise stop attacks.

Since the wall of fire is opaque, creatures inside can't be seen. Thus:

When you attack a target that you can’t see, you have disadvantage on the attack roll.

(PHB, p. 194)

This means:

• Melee and missile attackers make their attacks at disadvantage.

• Creatures on one side of the wall may roll a Dexterity(stealth) check to become hidden to foes on the other side of the wall, requiring attackers to guess which locations to target with their attacks.

• Nothing in the spell description specifically says that making attacks through the wall will inflict damage on the attacker.

• Nothing in the spell description specifically says that grappling or shoving are affected.

• However, the description of wall of flame does say that a creature who "enters the wall" takes damage. A DM could quite reasonably rule that grappling, shoving, and making melee attacks with some weapons (short ones) constitute "entering the wall", and impose damage in those cases.

• Any spells that target a creature or point that the caster can see cannot target inside the wall of fire.

• Spell effects extend through the wall (the description of wall of fire doesn't say that they don't), so spells that affect an area that originate from outside the wall can affect creatures inside it normally.

• Note that many spells that are attacks (for example, fire bolt), don't target a creature the caster can see, just one within range. So these spells can be used against creatures inside the wall, but, just like missile attacks, they will be made at disadvantage (or just miss outright).