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.
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grammar
Sentences ending in て
It is one of the most used forms to give commands, but in a semi-polite
manner. Maybe you have seen it with ください appended, which is used to increase
politeness, as adding "please" to a command. I have been told by my teachers
that it still is a command and not a request.

In your first example リー is telling (commanding) 山川 to advice her when coming
to her place. In the second example 山川 is telling (commanding) the other
person to bring 2 bags as big as the referenced. However, the て form that リー
is using is to join actions, in other words, リー is thanking the action
received.

In the Shin Chan example, the person is telling (commanding) other person to
bring a thermometer.
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In your examples, the 〜て form of the verb is to make it a command.

I'm not sure if you've already come across this grammar point before, but to
turn a verb into its command form, you convert the verb into the 〜て form and
add ください after it.

For example, 持つ (to hold) → 持って下さい (please hold ~)

In casual conversation, it's common to remove the ください, and simply leave the
verb in the 〜て form.

Your first example would translate to something like:

山川: Really? I'm glad, thanks!

りー: When you come, contact me!

EDIT: I should probably note that while I call this the command form, this is
a polite way of making requests (Removing ください will make the request less
polite). There is a verb form known as 命令形 which is a much more direct (more
impolite) way of commanding someone compared to the 〜て form.

ーーーーーーー

EDIT 2: (EXPLAINING THE SECOND CONVERSATION)

I'm not entirely sure if this is correct, but I would assume that the 〜て form
for 持って is a shortened version of 持ってる。

So the conversation goes something along the lines of...

リー: Heey, do you remember? Last April when I first came to Japan, Yamakawa-san
came to pick me up from the airport, right?

山川: Of course I remember! Rii-san, you're holding two bags THIS big!?

リー: Yeah, yeah. I can't carry it all by myself, so could you help me? You've
been very helpful!
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As many said, 来るとき、連絡してね and 体温計持ってきて are the command usage. But 大きなかばんを２つも持って
and 手伝ってもらって are just a conjunctive (or adverbial) usage.

  * 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。 Hey, do you remember? Yamakawa-san, you came to the airport to pick me up when I first came to Japan last April.
  * 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。 Of course I do, Lee-san. (You came) with as many as two this big bags.
  * 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。 Yeah, since I couldn't bring them all by myself, (I did them) with your help. I really appreciate it.
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