Hi, good questions! However, there will be cases that my translations are not appropriate depending on the situation. Please remember about that.
1. Thank you for calling technical support, how may I help you today?
テクニカルサポートにお電話いただき、まことにありがとうございます。本日[ほんじつ]はどういったご用件でしょうか / どういったことでお困りでしょうか。
2. Could you please connect the cable...
ケーブルを / ケーブルにおつなぎいただけますでしょうか。
*(using -te kudasai I suppose?) --> ~てください is the polite way but the honorific should be needed for customers.
3. Is there anything else I can do for you today?
何かほかにもお困りのことはございますでしょうか。
4. I will be more than glad/ happy to assist you…
何かお役に立てることがございましたら、どうぞおっしゃってください。
何かお役に立てることがあれば幸いでございます。
*I think it depends on the kind of support and the situation so I'm not very sure my translations are appropriate or not.
5. I’m sorry for the inconvenience that you have faced, what I can do for you is...
ご不便をおかけして申し訳ございません。本日(わたくしどもが)お手伝いできますのは / 本日わたくしどもがサポートさせていただけることは…でございます。
6. Let’s see what we can do to fix this...
ではどのように調整 / 修繕 / 改善 できるか拝見してみましょう。(the person who sees is the assistant)
… いっしょにごらんください。(the one who sees is the customer)
7. This will be fixed by the end of the weekend...
週末までには調整 / 修繕 / 改善 できるかと思います。
8. We’ll give you a call as soon as we've had an update…
(I don't understand this UPDATE for my English. Does it mean "the newest information", "the day we finish fixing (something) or other thing?)
*I wrote what the assistants say in general, however, I never worked as an assistant like that nor I'm not a professional teacher, so there might be errors in my Japanese. I hope other person reply to you. I also hope this was helpful.