How can I load lots of data at a time with Aidbox API?

You can use Bulk API for that

How to increase speed/performance for requests with complex parameters?

Create indexes for that.

Does Aidbox support 2-factor authentication?

Yes, please, check out the documentation.

How to create a SearchParameter for extensions on a property in a resource?

You can create custom search parameter for extensions and to made SearchParameter easier to write you also can define attribute

I’m building a request with revinclude. Is it possible to also apply a search for the include resource in that request?

In the FHIR Search API this is called chained-parameters. Read the details here and here

Aidbox appears to have a POST size limit of 20 MB by default. Is this configurable?

For sizable data, we recommend using Bulk API

I don't want to use BASIC authentication for security reasons. What can I use instead?

We suggest OAuth implementation in Client Credentials Grant. You need to use “Client Credentials Grant + Access Policy”:

My search doesn’t work with the full link to the resource included. Why is that?

An example of the query that doesn't work:
GET /DocumentReference?patient=https://my.aidbox.app/Patient/1234
It’s called an absolute reference. Aidbox doesn't work with absolute references because the logic can become very tricky since these references can point to external servers.
Use relative references instead: GET /DocumentReference?patient=Patient/1234