I don't work enough with the SDK enough to tell you anything about 10.0 vs 10.1 and compiling.
I can tell you that the underlying architecture for background processing changed in between 10.0 and 10.1. However as far as I understand it, this change should not have had any impact on a custom tool. If your tool works in desktop (foreground), then it should work in background (32bit).
The 816 error is fairly generic yes, but it basically means that the tool cannot be found/used (as you already know). Like I said, if the tool works in DT, it should work in BG. I've only ever seen 816 thrown in regards to "works in desktop, but fails in 64BIT- BG". Your best bet might be to contact TechSupport. They'll ask you to share your tool and see if it works for them.
To answer your third question, at 10.1 a new interface was added which allows you to force the tool to run in the foreground. You can read more about it here.
I realize you said you aren't going to worry about 64-bit BG right now but you found the right KB article which explains how to register your tool for 64bit. Because you're referencing that I assume you're working with C# or VB, not C++. If you haven't seen it, theres a little bit of developer information regarding 32/64bit here.
EDIT: Small code fragment for IGPToolBackground
public class CopyFeaturesFunction : IGPFunction2, IGPToolBackground
{
…… IGPFunction2 methods ….
public bool CanRunInBackground
{
get
{
return true;
}
set
{
return;
}
}
}
It is a replacement
ArcGIS Pro .NET SDK: The Road Ahead
–
SDK and API
No need for “external” PIAs (primary interop assembly) like with Arcobjects. Simply install ArcGIS Pro.
This is the fully fledged 64-bit version that will use more than 4GB of RAM for processing a long drawback of ESRI - welcome to the 21st century
Still limited
At Final Release: No APIs for: Facility Network Network Analysis
Labelling Data Reviewer APIs still undecided for: Data Management 3D
Analysis (TIN, LAS) Layout ArcGIS Pro .NET SDK
ArcGIS Pro API Workflows at 1.0, 1.1: 2D and 3D editing 2D and 3D Map
interactions (author, query, time, visualize) Content management and
sharing (online, portals, web maps) Fine grained APIs for Geometry and
Geodatabase Leverage .Net language features and async patterns
What is not supported in the API for Pro?:
Custom data source and feature class extensions Custom extensions
(edit, feature class, map) Custom Raster functions Custom GP functions
Standalone “ArcEngine” applications ArcGIS Pro .NET SDK: The Road
Ahead
source:
http://proceedings.esri.com/library/userconf/proc14/tech-workshops/tw_725.pdf
Best Answer
.Net 4 is supported by ArcGIS 10.1, but there is a little extra work you will need to do. http://resources.arcgis.com/en/help/arcobjects-net/conceptualhelp/index.html#//000100000nzv000000