[SalesForce] Inserting ‘Log a Call’ task using REST API

This is possible in java. I tried your code and it is working. As per my observation only thing which could be culprit here is the uri. Below is the code snippet where I have mentioned uri directly.

import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;

import org.apache.commons.httpclient.HttpClient;
import org.apache.commons.httpclient.methods.PostMethod;
import org.apache.commons.httpclient.methods.StringRequestEntity;
import org.json.simple.parser.JSONParser;
import org.json.simple.parser.ParseException;


public class Main {

        public static void main(String[] args) throws FileNotFoundException, IOException, ParseException {
            JSONParser parser = new JSONParser();

               // Parser parsing  JSON File having multiple records
               Object obj = parser.parse(new FileReader("JsonData.json"));

               HttpClient httpclient = new HttpClient();
               PostMethod mypost = new PostMethod("https://kmforce-dev-ed.my.salesforce.com/services/data/v34.0/composite/tree/Account/");
               String jsonText = obj.toString();
               System.out.println("\n_JSON TEXT=" + jsonText);
               mypost.setRequestHeader("Authorization", "OAuth " +"<myssessionid>"); 
               mypost.setRequestEntity(new StringRequestEntity(jsonText, "application/json", "UTF-8"));
               int response=httpclient.executeMethod(mypost);
               System.out.println(response);
               System.out.println("Response-->"+mypost.getResponseBodyAsString());
        }
}

Below is the output:

{
    "hasErrors": false,
    "results": [
        {
            "referenceId": "ref1",
            "id": "0019000001bslYJAAY"
        },
        {
            "referenceId": "ref2",
            "id": "0019000001bslYKAAY"
        },
        {
            "referenceId": "ref3",
            "id": "0019000001bslYLAAY"
        },
        {
            "referenceId": "ref4",
            "id": "0019000001bslYMAAY"
        }
    ]
}

Let me know if you face any issue here.

1. The picklist-values-per-record-type issue is an old problem to which there's only recently a semi-decent solution. There's a number of ways to approach this, and if you're already using the API, the new UI API is probably the best way to go.
2. You can use a variety of routes to access record type availability. Issuing a Describe API call to, e.g., /services/data/v43.0/sobjects/Opportunity/describe has keys available (for the running user's Profile) and defaultRecordTypeMappingunder each of the entries in the recordTypeInfos. The Metadata API also permits you to get record type availability and default record type information at the Profile level, but it's non-trivial. The Tooling API can theoretically also yield this information, although I wasn't immediately able to get the record type visibilities that way in Workbench. (Looks like others have had that issue too).
3. This is the easiest, but it has nothing to do with the page layout. Just set the field RecordTypeId on the Task you're inserting to the Id of the desired record type.

Note that if the Profile has exactly one custom record type associated with it, that will be used automatically.