Congratulations for being in such high demand :-)
I don't think you're going to be able to charge an hourly rate if the bulk of your correspondence is via email. It's going to cause more problems then it's worth.
Airpair.com provides 24 hour access to an expert, priced based upon the complexity of the problem (a proxy to how much work is actually required on your part). They also calibrate the expert assignments, making sure the more advanced professionals get the more difficult problems. This could be a possibility for you - unless you want to continue handling all requests yourself.
I wonder if you could use Slack as the primary platform for orchestrating the transaction.
1. User hits your website, becomes convinced, submits the inquiry, and deposits money into his account. Managing a balance adds complexity, but you always have a better chance of getting more money with one transaction versus smaller ones.
2. Upon payment (let's use Stripe, because that integrates with Slack) they are issued an invitation to your Slack instance. A private channel is also created that they will be restricted to. The private channel should correspond to some sort of customer ID - because you will want to use the same channel throughout the lifetime of the customer to maintain continuity,
3. Optionally you may also create a tracking ticket using something that integrates with Slack and Stripe. The ticket contains the stripe transaction information, the original inquiry, key info on the customer, and hopefully links to the customer's past interactions.
4. The advantage of Slack is all of the built in integrations. Most of your back and forth can be via the persistent chat - and you can have a bot reminding the user not to expect instantaneous responses. But if you do need to do screen sharing, video conferencing, etc...there is no leaving the environment - and it all gets tracked.
5. You may be able to fairly track time if Slack can measure your time in the channel and decrement the available funds accordingly. Just make sure you're only in the channel when you're either actively communicating or researching on the user's behalf.
6. Once the user's account hits a predefined threshold, issue a warning and suggest that they replenish their account in order to continue the conversation. Clicking the link or interacting with the bot would replenish the account via a charge out to Stripe. Tracking ticket is updated.
7. When the user shifts topics (after you satisfy one request and he moves onto the next), give a bot a command that updates the ticket by closing the last request and opening a new one. Of course we're getting a bit nuts with this level of tracking - but I think it will be valuable to keep them all separate as I am sure you get a ton of repeat requests. Hopefully you'd be able to trigger an email to the user with a full transcript of the exchange.
8. Eventually the user will be done asking questions and may likely have a balance. Even if the balance is zero, retain the private channel. It serves as a searchable log of the interaction and is a mechanism for the user to come back and reengage with you.
9. You may optionally add some public channels covering the topics you provide advice on. This may be useful to keep your users occupied while you answer questions as your day progresses.
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It's important to note that this is only worthwhile if the exchange between you and the customer is more discussion-oriented versus one-off answers to specific questions. Simpler interactions would remove the value Slack adds - and in that case I would fall back to a ticketing system you're comfortable with that has hooks into the payment engine you're using.
:-)