Yes, they are exactly the same.

They remain a unit until their next Movement phase, when they must separate if able to do so (i.e. if they are not fleeing, rallying or subject to another form of compulsory movement). Note that, if they are fleeing, they are allowed to take a single rallying test as normal, as they are still a single unit.

Well, we find that there are two ways people use to solve this issue, and both are equally acceptable: 1) placing the character inside the unit and 2) placing the character next to the unit (in base contact with the flank of the unit's first rank).

Let's look at these two solutions in a bit more detail:

1. Placing the character inside the unit. This works best when the character's base is larger than those of the unit's and is a multiple of them – for example a model mounted on a 40mm square base inside a unit of models with 20mm square bases, or a character on a 50mm square base inside a unit of models with 25mm square bases or a cavalry unit. In these cases, the character will fit in snugly and look and feel 'right'. It will also displace a number of models, which are placed in the rearmost rank as normal.
   
   When working out the unit's rank bonus, the character counts as the same number of models it has displaced (normally four infantrymen or two cavalrymen).
   
   This can also be done if the character's base is smaller than those of the unit he's joining or it does not fit neatly with those of the unit he joins (the most classic case being that of a character mounted on a horse joining an infantry unit with 20mm square bases).
   
   In this case, however, you will have to slightly 'fudge' the unit's formation (that's a technical term, honest!). In other words, just pretend the character's base is the same as the other models in the unit (for foot characters), or equal to a multiple of the bases of the other models in the unit (for mounted or large models joining units of smaller bases...). This will make the formation a little messier, but it's normally quite self-evident.

2. Placing the character next to the unit. This approach is not strictly speaking 'by the rules', but it's very simple (particularly for characters with bases that are different from the models in the unit they're joining) and thus a favourite in friendly games, but we really cannot recommend it in tournaments and other competitive games...
   
   The character is simply placed in base contact with the flank of the unit, facing in the same direction as the rest of the unit and flush with the unit's front. It is treated as 'inside' the unit from all points of view, but the unit's formation is not disrupted by the character's differently-shaped base.
   
   This works fine except if the unit is then charged in the flank where the character is, but with a bit of flexibility, and trying to apply the principles presented in "Close Combat & Incomplete Ranks" on page 36, you should be able to solve this easily.

Common sense would imply that the character counts as moving, but the unit does not.

They will target the character's unit, and if the unit panics and flees, it will take the character with them.

The excess wounds caused are those after the multiplying, so in the example above the combat resolution bonus from overkill would be the full +6.

Characters engaged in a challenge are still in base to base contact with other friendly/enemy models. Thus the challenge participants may be affected by magic items/abilities originating from outside the combat (and vice versa).

Impact hits will affect the unit and not the character as normal if there are more than 5 models in the unit (as described on page 77), but will affect the character if he is the only model left when the chariot charges in. If there are less than 5 models the impact hits are distributed as shooting hits.

No, they cannot, just like the crew of the chariot cannot attack them.

No, the Champion cannot be harmed if he's fighting in a challenge.

No, if the unit he is with is not subject to Fear/Terror, he does not have to test either, as he is with them when the charge is declared.

Well, these situations are not covered by the rules, so we'll make some recommendations for you. If he fails a Fear test, we think it's better if he suffers the effects of the failed test, except that he can stay and move with the unit in the unit moves later in the phase. If he fails a Terror test, we think it's better if he flees on his own and leaves the unit behind.

No, they are characters and not monsters, even though they share some rules with monsters (normally this is limited to the way they move and their inability to join units).

The unit must use the same type of movement (i.e. ground movement or flying movement), so the Black Knights in the example may charge up to 12", as the unit itself cannot fly. I chose this interpretation because the other would have implied having to consider what happens if such unit charged through terrain or, God forbid, through a wood!

At the end of the phase, the monster is moved outside the unit, in base contact with it, as close as possible to the position it was occupying when ridden.

Because the rider and mount can be hit separately, any saves and other special rules of the character (including those from magic items, spells, etc.) are not passed to its mount, and viceversa. There are a few exceptions to this rule however, when such rules do apply to both rider and mount: Psychology rules (see rule book, page 79), rules that the character would confer to a unit it joins (like Magic Resistance), or if the rule itself specifies otherwise (certain magic items, the Blessing of the Lady, etc.).

It is easier if this works the same as the Immune to Psychology rule, so the character would lose the benefits of this rule as long as he stays with the unit.

No, in the same way as if the character was Immune to Psychology, for example.