Data Structure Logic:

Hashmaps are used to access the content via its key.

In this sample, I guess you are scanning 'tuples' one by one and each tuple is different from other, so you won't need to access an old tuple via it's key.

So here, I suggest you to create the model class suitable for the key-value pairs and store them in a list. You can push that list when you are done.

Sample model for tuple:

public class KeyValuePair {
    private String key;
    private String value;
    public KeyValuePair(String key, String value) {
        this.key = key;
        this.value = value;
    }
}

Sample List to store:

List<KeyValuePair> keyValuePairList = new ArrayList<>();
keyValuePairList.add(new KeyValuePair("0", "816444014066"));
keyValuePairList.add(new KeyValuePair("1", "747083010945"));
keyValuePairList.add(new KeyValuePair("2", "816444010969"));

Storing:

If you cannot pass the data between activities, check out SQLite. You can store the data in SQLite and fetch it when needed. It is an offline database works on Android devices. You can delete the data when pushed to upstream.

Edit:

If the keys are the orders, you can simply use a String List like this:

List<String> keyValuePairList = new ArrayList<>();
keyValuePairList.add("816444014066");
keyValuePairList.add("747083010945");
keyValuePairList.add("816444010969");

You can keep the Map. The Map is designed to be looked up by key so I suggest you have a list of shuffled keys.

public Map<Integer, String> getQuestionOptionsMap() {
    Map<Integer, String> map = new HashMap<>();
    String[] answers = {null, answer1, answer2, answer3, answer4};
    for (int i = 1; i < answers.length; i++)
        if (answers[i] != null)
            map.put(i, answers[i]);
    List<Integer> order = new ArrayList<>(map.keySet());
    Collections.shuffle(order);
    Map<Integer, String> shuffled = new LinkedHashMap<>();
    for (Integer key : order)
        shuffled.put(key, map.get(key));
    return shuffled;
}

First, I'd like to point out that if are you storing this from your program and reading it back from a file later, you'd be better off Serializing it. But since you're being given this file, let's move on.

Infinite Loops

It's generally discouraged to use while(true) loops. It's not particularly bad (in fact it's sometimes better), but it's always best to look for an alternative for readability sake. Instead, let's use while ((line = reader.readLine()) != null).

Initializing variables as null

I also see you initialized a few variables as null, but that isn't necessary since it's implied for classes. (Also note primitive variables are implied as 0 during initialization).

Using the output format to your advantage

You also have a cut variable that gets the index of the tokens, and then skips the line if it's blank. However, we know that the output is always going to be grouped into three lines, and we always know what those lines are going to be. We also know the data you really want is going to be the first three characters for the name, and everything after the first five characters for the tokens. Let's use that and make everything clear!

Putting it all together

String response = restTemplate.getForObject(url, String.class);
BufferedReader reader = new BufferedReader(new StringReader(response));
NameMetrics current;
Map<String, String> metricsHolder;
List<NameMetrics> result = new ArrayList<>();

String name, load, peak;

String nameline;
while ((nameline = reader.readLine()) != null) {
    name = nameLine.substring(0, 3);
    load = reader.readLine().substring(5);
    peak = reader.readLine().substring(5);
    reader.readline(); //eat empty line

    current = new NameMetrics();
    metricsHolder = new HashMap<>();    //No need to repeat the stuff inside the diamond operator
    metricsHolder.put("Load", load);
    metricsHolder.put("Peak", peak);

    current.setName(name);
    current.setMetrics(metricsHolder);
    result.add(current);
}

Handling varying keys and names

When there are a variable number of keys, we still know that they're all generally in the same format of the name followed dashes, then a list of key:value lines, then a blank line. We can still use this to our advantage as follows:

String name;
String[] mapValues;

final int KEY = 0, VALUE = 1;
String nameline, keyLine;
while ((nameline = reader.readLine().trim()) != null) {    //Loop until the end of the file
    name = nameLine.substring(0, nameLine.indexOf("-") + 1); //Plus because substring excludes last index

    metricsHolder = new HashMap<>();    //No need to repeat the stuff inside the diamond operator
    while (!(keyLine = reader.readLine().trim()).isEmpty()) {   //Loop until block ends with an empty line
        mapValues = reader.readLine().split(":");   //Split key:value format
        metricsHolder.put(mapValues[KEY], mapValues[VALUE]);
    }

    current = new NameMetrics();
    current.setName(name);
    current.setMetrics(metricsHolder);
    result.add(current);
}

Use String.split:

String[] kvPairs = "key1=value1;key2=value2;key3=value3".split(";");

This will give you an array kvPairs that contains these elements:

key1=value1
key2=value2
key3=value3

Iterate over these and split them, too:

for(String kvPair: kvPairs) {
   String[] kv = kvPair.split("=");
   String key = kv[0];
   String value = kv[1];

   // Now do with key whatever you want with key and value...
   if(key.equals("specialkey")) {
       // Do something with value if the key is "specialvalue"...
   }
}

Here's a relatively neat way:

var map = IntStream.range(0, list.size() / 2).boxed()
    .collect(Collectors.toMap(i -> list.get(i * 2), i -> list.get(i * 2 + 1)));

you can add a number to say how many splits you want just add a 2 to split

import java.util.HashMap;

public class HelloWorld{

     public static void main(String []args){
        HashMap<String, String> holder = new HashMap();
        String payload = "cossn=0, abc=hello/=world";
        String[] keyVals = payload.split(", ");
        for(String keyVal:keyVals)
        {
          String[] parts = keyVal.split("=",2);
          holder.put(parts[0],parts[1]);
        }

     }
}

Map<KeyType,ValueType> which is implemented by HashMap<KeyType, ValueType> and TreeMap<KeyType, ValueType>, among others -- HashMap is unordered and TreeMap is ordered.

Other useful Maps are LinkedHashMap which is like HashMap but iterates in insertion order, and com.google.common.collect.Maps in Guava which has a bunch of utility methods, and com.google.common.collect.ImmutableMap which is an immutable map implementation.

For your key corresponding to an array of strings, you might want to look at a Multimap which is a map with multiple values for a given key.