The Gemarahs in Kereisos and Horayos and Shulchan Aruch, list five items that Abaye says one should eat (or see) on Rosh Hashanah. Since he did not say etc. why did he limit his statement to just these 5 items, and secondly, is there any significance in the order he listed them: kara (squash), rubia (type of peas), karti (type of long green onion), silka (beet leaves) and t'marim (dates)?

There are two classic explanations: Rashi writes that these foods are either sweet or grow quickly, symbolizing that we should merit those blessings in our lives, while the Geonim write that their names in Hebrew/Aramaic symbolize those key words in the brachot which we wish to bless the New Year. The Abudraham, Tur and others mention that these 5 are the source of the custom to add additional foods which add further blessings, where the idea (and the answer to your first question) is clear: the more blessings at the beginning of the New Year, the better! The order mentioned there isn't significant, for the achronim davka change the order. According to the Geonic explanation, it makes more sense that our enemies should first leave us (yistalku-silka), and then be done with (yikartu-karti) and then be finished (yitamu-tamar), for if (according to the Talmud's order) they are "done", they no longer need to "leave us". Also, according to the instruction of first "sur mera" (do away with negative), and then "aseh tov" (bring the positive, Tehilim 34, 15), we should first address the more urgent life-saving problems (the aforementioned 3): our enemies who wish to obliterate us- which even supersedes the prohibitions on Shabbat, and afterwards bless that our positive merits should be read before You (yikri'u-kara), and should be multiplied (sheyirbu-rubia).