Answer: Yes. The Orchos Chaim holds that one has not fulfilled the mitzva if both are held in the same hand. While the Taz disagrees, the Mishna Berura concludes that one should follow the stringent opinion and shake them again without a beracha.
Answer: Shulchan Aruch 651:5 writes that one should first lift the lulav and only afterwards the esrog and the Chayei Adam concurs. However the Magen Avraham writes that the esrog should be picked up first and put down last. The custom and majority of Poskim follows the first opinion.
It is accepted by the majority of the Poskim that the esrog is the only of the 4 minim to have Kedushas Shmitta issues. The other three may be wrapped and thrown away, but should not be placed in a dirty garbage can unwrapped as it is a disgrace to the mitzva. The minhag, however, is to use them for another mitzva such as burning the chometz. The esrog must be disposed of before Shvat either by leaving it to rot naturally or by placing it in the garbage at least double wrapped.
The first priority is that the esrog should be clean from noticeable blemishes, especially the top part. After that it should be a good size and yellowish color as well as a shape like a "tower". The actual determination by the Rav whether the esrog is "mehudar" or just "kosher" is a subjective assessment of all the above factors. For more information see the article at Revach.net about choosing an esrog.