The "Ke’ara" is the plate which we keep on the table throughout the Seder, and it contains all the special foods eaten at the Seder. One should ensure that the plate contains all the foods throughout the Seder. Meaning, after one partakes of the Karpas, for example, he should keep some leftover Karpas on the Ke’ara. Even though one has already passed that stage of the Seder, it is important that the Ke’ara has on it all the foods, even the Karpas.
The salt water, into which one dips the Karpas, does not have to be on the Ke’ara.
The custom in our community is to eat specifically celery, which is the food denoted by the word "Karpas." This is the custom that one should follow, as the word "Karpas" alludes to the "Perech Samech" – the backbreaking labor endured by the 600,000 Israelite men in Egypt. Furthermore, Rav Haim Vital (1543-1620) taught that the according to Kabbalah, the numerical value of the word "Karpas" (360) is very significant and alludes to different Names of the Almighty.
Before eating the Karpas, we wash our hands the way we do before eating bread. That is, we pour water three times on the right hand and then three times on the left hand. The only difference is that no Beracha is recited upon this washing. It should be noted that the requirement to wash before Karpas relates to a general Halacha that is not connected specifically to Pesah. All year round, before one eats a fruit or vegetable that is moistened with a liquid, he must first wash his hands, without a Beracha. For example, people generally wash grapes and apples before eating them. Assuming the fruit is still wet when one eats it, he must first perform Netilat Yadayim, without a Beracha, before eating. We therefore wash our hands before eating the Karpas which is dipped in salt water. The Kaf Ha’haim laments the fact that most people are unaware of this Halacha, and they wash Netilat Yadayim before Karpas but not before eating wet foods other times during the year.
We dip the Karpas in salt water in order to do something unusual that will arouse the children’s curiosity at the Seder. Normally, at that point in the meal we eat bread. When they see that we instead dip celery in salt water, they will find this unusual and ask questions. There are also many Kabbalistic concepts underlying the dipping of Karpas, so one must ensure to properly observe this and all customs at the Seder in accordance with tradition.
Several works mention the importance of saying or singing the names of the various stages of the Seder (Kadesh, U’rhatz, Karpas, Yahatz, etc.). Before one begins each stage, he should say or sing all the stages starting from Kadesh, and then stop upon reaching the current stage. Before Karpas, for example, one would recite, "Kadesh, U’rhatz, Karpas." The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) expounded upon the Kabbalistic significance of each of these words. One must not belittle or neglect these or other traditional customs, as they are all based upon profound, underlying meaning and wisdom.
Summary: One washes Netilat Yadayim without a Beracha before Karpas and anytime he prepares to eat a food that is wet. One should use specifically celery for Karpas. After eating the Karpas, one should still make sure that some Karpas remains on the Seder plate. It is proper before each stage of the Seder to state all the stages from Kadesh until the current stage.