grammar - Sentences ending in て

While in Japan I came across the Kinokuniya mega bookshop, which I fell in love with. I went to one of the top floors, started browsing the Japanese learning books, and found a couple focused on listening and speaking, two of my weakest points, and completely in Japanese, which is another plus. In general they're not too difficult, as I got the ones most suited to my level, but now and then I come across certain things I haven't seen before. One of them is sentences ending in て

An example, from a listening exercise (dialogue between two international students and their host, before going back to their country.)

• 山川：本当？うれしい、ありがとう。


• リー：来るとき、連絡してね

Another example:

• リー：ねえ、覚えてる？去年の４月、私が初めて日本へ来た時、山川さんに空港までむかえにきてもらったね。


• 山川：もちろん覚えてるよ。リーさん、こんなに大きなかばんを２つも持って。


• リー：そうそう。一人で運べないから、山川さんに手伝ってもらって。本当に助かった。

I have also found examples of it on a Shin Chan volume I'm reading, for example:

• おやつ食べたら体温計持ってきて

So, what's the purpose of finishing verbs in て at the end of the sentence?

EDIT: I understand the て (order) from the first conversation and the Shin Chan extract. Only the second one is left, which from context I understand is not an order.