WebJobs running in the free tier are limited to 20 minutes. Otherwise, they can run on demand, continuously (as long as the corresponding Web App Service is running), or on a schedule. Right now you have to use the old Azure UI (manage.windowsazure.com) to set up a scheduled job.
You can create and deploy WebJobs from Visual Studio, using continuous delivery, or from the (new) Azure Portal (portal.azure.com). For a Mobile App Service, you would add a WebJob at Mobile App Code > All Settings > WebJobs > Add. (Azure Batch Services are different from WebJobs, primarily in scale, though the similarities can be confusing at first glance.)
Azure API App Services
As I described above for Connectors, Azure API App Services use Swagger and REST as pluggable interfaces and JSON as the interservice data format. I built and deployed a sample API App Service (Figure 5) in C# from Visual Studio, following an Azure tutorial, without any problems to speak of once I had upgraded my Azure SDK to the latest version. You can use ASP.Net, Java, PHP, Node.js, or Python for your API Apps. Ruby is not yet supported for this, though it is supported elsewhere in Azure.
Right now, C# is the only language for which the Azure SDK adds an API App Service project to Visual Studio 2013. I expect that to change by the time Azure API App Services come out of preview.
The API App host takes care of managing authentication for the app, which saves you the headache of implementing it yourself. Otherwise, you can build your own REST interface on a Web App Service if you wish. I'm not clear on whether you can expose a REST/Swagger API to composition in Logic App Services if it's built on a plain Web App Service. You clearly can if it's running as an API App Service.
Azure Logic App Services
As I implied, Logic App Services can visually compose Connectors and other API Apps into a business process (Figure 6). In the example shown, I set up a process to search Twitter for tweets about InfoWorld hourly, using my Twitter account, and to save any tweets found into my Dropbox, using my Dropbox account. I had to authorize both Connector apps to use my accounts. Running the app manually, for testing, did in fact create a file from a tweet and save it in my Dropbox. The code view of the Logic App Services blade shows the generated XML for the business process.
All Logic App Services start with trigger. In the sample in Figure 6, I scheduled the service to run hourly. I could as easily define the trigger as an event from another service, such as a SQL Server insert or update trigger.
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