Are monster cards allowed to change positions on the field?

Answer 1

Yes, they are allowed to change battle position, although this may only happen once per turn (unless an effect otherwise allows it). They may be flip-summoned, but ince face-up they remain so; if summoned face-up then they cannot be later flipped down, again unless an effect allows them to do so.

If you are asking if they can move into a different monster slot (ie, a monster is placed in slot 1, and you are asking if it can be moved across into slot 2), then no, this can't be done normally. Some cards in CDIP had effects that let them move into different slots, and gain bonuses against monsters in the same column on the opponent's side of the field.

Answer 2

As long as that monster has not declared an attack, or manually changed battle position already that turn, or was summoned on the turn, then yes.

If you'd just summoned the monster, or declared an attack with it, or changed it from defence to attack position, then if Book of Moon or Enemy Controller is used, then you can't manually change that monster's position until your next turn.

Answer 3

Here is a list of Spell cards that will let you change monsters from defense to attack position on your side of the field:

Earthquake

Book of Eclipse

Bubble Shuffle

Curse of Fiend

Level Limit - Area B

Karakuri Cash Cache

Karakuri Gold Dust

Spider Web

Spellbook of Fate

Spark Blaster

Spirit Burner

Stumbling