Does Esri allow vendors to create an ArcGIS Server alternative that is compatible with their tools?

Yes! Esri is committed to Open standards. In 2010 Esri open sourced the REST protocol (Geoservices Rest Specification) that is used to communicate with all of their SDK’s and client software.

From the specification introduction:

Why Implement the GeoServices REST Specification?

The GeoServices REST Specification offers a simple way for applications to request map, feature, attribute, and image information from a GIS server. Developers who adopt the GeoServices REST Specification are choosing a proven implementation that has been widely deployed and exercised in the field and that exposes server-side resources to a broad range of clients and applications. They are also choosing a JSON-based, REST-ful specification that will make the server instantly usable by thousands of developers working in popular client-side development environments with the ArcGIS web mapping APIs for JavaScript™, Flex™, Silverlight®, iOS®, and Android™, all of which are powered by the GeoServices REST Specification.

See this blog post for more information on the Geoservices REST specification.

Can I use Blue Spatial Server with ESRI SDKs and ArcGIS Desktop?

ArcGIS Desktop
If you have purchased ArcGIS Desktop you may use it with Blue Spatial without using another license.

According to the Esri product specific terms line 64: You can use Esri SDKs as long as you are also using another Esri product e.g., geocoding or base maps.

See further discussion on the blog.

As an alternative you could use Leaflet with Open Street Map basemaps and Blue Spatial.

What kind of performance can I expect?

Our load test shows .2 to .3 second response times at significant loads with low utilization for a point layer. We can help you setup a similar load test for your data.

For layers with large geometries our Map Service can generate tile caches which respond similarly.

Which features do you not implement?

Blue Spatial is almost at feature parity with ArcGIS Server for Map Services and Feature Services. The only features not included at this time are the following: 3D and Disconnected Editing. See the ArcGIS Server Feature Comparison.

Do you charge for cores? Named user instances?

No. We have per server flat pricing.

How is your cloud deployment model different than Esri's?

Esri cloud deployment follows the Infrastructure as a Service pattern. With Esri you deploy virtual machines into the cloud and you must manage those VMs at the OS level.

Blue Spatial is more of a Platform as a Service model. You can deploy to Azure Web Apps instances that do not require any OS level management. Azure Web Apps can be easily configured for automatic scalability.