Hi:
Yes, Google has lots of confusingly-similar service names, and, to make matters worse, the function of "Google Wallet" changed completely.
Google Wallet: was an app to store various credit/debit cards and use your phone to pay, similar to "Samsung Pay", and whatever Apple calls their service. Google moved that function over to a newer app called "Android Pay". Google Wallet is now only used as a peer-to-peer money transfer tool. For example, you and I go out to lunch, you pay the bill, and then I use Google Wallet to send you my portion of the bill. It's like "Paypal Me".
Google Payments: this is Google's overall system used to pay Google for things, like music or movies or whatever. It can store multiple credit or debit cards, and you can select which one you want to use for a purchase. When you pay the $10 number change fee, this is how you will pay for it.
G-Suite is the new name for "Google Apps", which is Google's cloud-hosted suite of productivity applications, including word processing, spreadsheet and graphics. Think of it as Google's competition to MS Office.
If you don't want Google to store your credit card number, just delete it after you make the payment. Paypal is not an option.
With regard to the "keep your old number" option, you have 90 days from the date you change your number. The intent of that time interval is for people who plan to abandon the old number, to give them time to let all their contacts know that they've changed their number. If you don't pay that second fee, your number will disappear on day 91.
With regard to Simonics, all you need to do is to click the "deauthorize..." and then "reauthorize" links on his web site.