Unless otherwise forbidden by the rules (or in a Q&A), a character can always join a unit, regardless of their mount. A unit always moves at the speed of its slowest model, so if a character on foot joins a cavalry unit, they would move at the character's Movement rate. A character only gains "Look out, Sir!" if part of a unit of "similar sized models". The same applies to targeting characters within 5" of a unit. This does not change, whether they are actually within the unit or not. So a character on a horse can still be picked out if he joins a unit of infantry, for example. For these purposes, a character on a monster base (40mm or larger) counts as being bigger than normal cavalry. A character in a chariot cannot join a unit unless it is a chariot unit, and only a character in a chariot may join with another chariot or chariot unit. A character takes up the space of an equivalent number of rank and file models as their base fills. For example, a cavalry model would take up one space in the first rank and one space in the second rank of an infantry unit. These ranks still count as complete if four "spaces" wide or more, whether that space is filled with a rank and file model or a character. See the diagrams on page 113 of Warhammer Chronicles 2004 for examples. View online.

They can be freely charged.

They still have it (and still count as a unit of two models). If they decide to split up, you can choose which character retains the captured standard.

Except where noted, a Champion is treated as a rank and file model.

If there is space in the single rank for him to be positioned where he cannot fight, he may refuse the challenge. If the entire rank is engaged, he cannot avoid the challenge, just as if he were on his own (the challenger hunts him down like the cowardly dog he is!).

If there is a lone character involved in a combat (even a multiple combat where there are more characters involved), the owning player's side can never refuse a challenge.

Number of Wounds doesn't matter, because a character with 3 wounds reduced to one, or a Champion starting with one wound is all the same for purposes of multiple wound weapons in a challenge. When attacks are allocated (when excess wounds may be wasted, unlike attacks against rank and file) then each unsaved wound is multiplied. In this case, you can tear the Champion apart and get your +6 combat resolution (1 Wound plus up to 5 overkill), even though he has only one wound.

The models engaged in the challenge are considered to be in single combat during the challenge and may not attack or be attacked by models outside of the challenge. Therefore the weapon / item only affects the opposing model in the challenge (and the bearer if it would normally affect him).

Yes. Simply because a single model is engaged in a challenge and cannot be attacked by a model outside of a challenge, does not mean that it is impossible for an additional enemy unit to charge the single model. The enemy may wish to do this to add ranks, Unit Strength, the flanking bonus, etc. to the overall combat result.