I'd like to speak to Mr. Carter ...
In this situation, "I'd like" + infinitive is a polite way of saying "I want" + infinitive. It is used for making polite requests.
Other examples are:
I'd like to arrange a meeting for next week.
I'd like to talk to you about last month's figures.
I'd like to speak to Mr. Carter if he's available, please.
This part of the sentence is used to check if Mr. Carter is busy or out of the office.
Other examples are:
If you're available next Friday morning, could we have a meeting?
If you're available , we'd like you to start work from April 1st.
Mr. Carter placed an order with us...
Basically, this means that Mr. Carter ordered some products. In this situation, when we want to use "order" as a noun, the verb to use with it is "(to) place".
Other examples are:
I'd like to place an order for 1000 leadframes.
I placed an order with them three weeks ago but they still haven't delivered the goods.
...I have a few questions for him regarding shipment.
In this sentence, "regarding" is used as a more formal way of saying "about".
Other examples are:
I need to talk to the client regarding the new contract.
Regarding the results of the experiment, I don't think the data is reliable.
I'd like him to call me back...
"I'd like him/her" + infinitive is used to make a polite indirect request. Ms. Imai wants Mr. Gibson to ask Mr. Carter , who is out of the office, to return her call.
More examples:
Mr. Tanaka, when Ms. Fujita returns, I'd like her to come to my office.
I'd like Mr. Shimizu to fax the latest sales figures, please.
I'd like him to call me back as soon as possible.
If you want something done quickly, you can say this politely by using "as soon as possible" at the end of a sentence.
Other examples are:
I'd like you to send the data as soon as possible.
I think we should leave as soon as possible.
Could I have your phone number, please?
"Could I" + infinitive is used in this sentence to ask permission for something. You can use "can I" in the same way, but "could I" is more polite. See also "May I" below.
Other examples are:
Could I have your name, please?
Could I ask you to fax your report tomorrow?
This is used to check important information, such as names and telephone numbers. When you say the sentence, remember that the intonation rises at the end.
"So that's..." is used here to confirm that the information received is correct.
And may I have your name, please?
This is used in the same way as "Could I" + infinitive (see above), but in a more formal manner.
Other examples are:
May I have your e-mail address?
May I ask how much your computer cost?
Let me just repeat
your message.
In this sentence, Mr. Gibson wants to confirm that he understands Ms. Imai's message by reading it back to her.