Kindly see answers inline:
Which API should they choose to buy a subscription for? I guess Bing Custom Search API? Could they also choose Bing Web Search API? What's the difference, in non-technical terms? They only want an internal search function for their website.
You are correct. They need the Bing Custom Search API.
If our clients choose Bing Custom Search API they can only choose the S1 pricing plan, right. When I clicked around in a test account I was asked if I wanted to upgrade it to a Pay-as-you-go-account, which I did. Is there a difference between S1 and Pay-as-you-go, or is Pay-as-you-go just the opposite of a free trial subscription (in other words S1, S2 etc. are all Pay-as-you-go plans)?
Pricing information for Custom Search is here. S1 tier with 100 calls per second is what you need. Just a cautionary note here: Bing Web Search also has S1 tier but that has 250 calls per second and you don't want that. Pay-as-you-go is just a methodology being used for billing. Free trial is available only for 30 days and you can get keys from here. Whereas, you can purchase subscription for custom search from here.
If our clients upgrade their free trial subscription before then end of the first 30 days, will the subscription remain free until the end of the 30 days, or will they be charged right away? In other words, is it beneficial to the client to wait 30 days with upgrading the subscription plan?
The free trial subscription will be free for 30 days with limited call volume. This will not be charged. Paid subscription has different keys compared to free trial keys and usage of paid subscription keys will incur charges straight away.
Will the subscription keys remain the same for the same resource after the subscription has been upgraded from free to paid tier? In other words, can our web developers implement the free tier keys right away on the sites and not have to change them when the client upgrades their subscription to paid tier?
No, subscription keys are different for free trial and paid subscription. When you buy paid subscription, you need to change keys from free trial keys to paid subscription keys. For implementation details, you can check out this blog.
