The effect "● Shuffle 1 card from your opponent's hand (at random) or their side of the field into the Deck, then this card gains 100 ATK." does not target. (Shuffling 1 card into the Deck and increasing the ATK of "Dinomist Rex" are not considered to happen simultaneously.)
Each time "Dinomist Rex" attacks, its effect can be activated at the end of the Damage Step. If "Dinomist Rex" attacks in a row using its own effect, the effect of "Dinomist Rex" can be activated again at the end of the Damage Step. If you do, you can choose the effect "● This card can attack an opponent's monster again in a row, and if it attacks a Defense Position monster, inflict piercingbattle damage to your opponent." again.