We generally use the Change Privilege geoprocessing tool from the Toolbox, to give editing/viewing rights over a particular dataset.
Since there are so many supporting tables that are used while editing, I would suggest that you use this GP tool to set the required privilege.
As pointed out by Alex in a comment,
arcpy.env.packageWorkspace
is an expected update in your script if you look at where the GP Service is deployed. The packageWorkspace is the directory itself. When publishing a service that uses SDE data, and you've referenced the SDE database in your datastore, a copy of the connection file (.sde) gets moved into the directory and your script is updated to reference that. This is expected and how its supposed to work.
You're right about "generally" speaking per your link different versions will work together:
According to this link I should however be able to publish form 10.2.2 to 10.1 without too many problems.
I do want to point out times though this wont be true. There have been a few tools which have been enhanced between 10.1 and 10.2.2. These tools can only be publish when each product is at the same version. (this is just a general note, this point wont be your issue).
There have also been some bugs fixed between the versions, mostly related to script update and data paths. Again, generally speaking most should work, but there could be cases that they dont. I dont have enough information about your workflow to say that updating Server to 10.2.2 will solve the problem (so I'm not suggesting that).
Your best test is with ArcMap, to navigate down into the v101 folder and find the .RLT (result) file. Drag that into your map. Open the Results window and under the "shared" node, run that. Thats basically like running the service, but running it locally, not via ArcGIS Server. If that works, its Server having problems connecting to SDE. If it doesn't work, its not resolving the path to your SDE connection file and I'd sort of expect the version differences perhaps messing something up. The easy "test" to prove that would be to hack the script in that directory to point to the connection file explicitly.
Best Answer
It's not strictly read-only access. You can load data via copy/paste and import into a SQLite or SpatiaLite database. Tables and feature classes can be appended to using append or Load Data.
You can edit the SQLite/SpatiaLite data via code as Insert/Update/Delete are supported.
What you can't do is open an edit session in ArcMap.