What would be the appropriate word for worrylessly in this following context "I shall prefer him for this task. And [worrylessly] expect an amazing result.
The correct word would only have one y: "worrilessly." While "worrilessly" may not be in your dictionary, "worriless" certainly is. It's proforma that an adverb can be made of any adjective that was original formed from a verb by adding "ly." Since the adjective "worriless" was formed from the verb "worry," you can add "ly" and have it be a legitimate adverb.
The dictionary doesn't list all modifiers that are formed from adding suffixes to verbs. For example, you can add "ed" and "ing" to any verb to form a modifier, but the large majority of dictionary definitions for verbs don't list verb+"ing" or verb+"ed" as modifiers, just like many don't list formations from adding "ly." Only those most commonly used are in the dictionary.
Considering the context in the question, there are a few adverbs that could be used in the context. For example:
In an unworried way; calmly, without anxiety.
The adverb calmly could be a candidate.
Now, the issue is how the adverb unworriedly could be used idiomatically in your context. This adverb is not broadly used even though it could get the message across. If your context means you are without worry in terms of his performing his task successfully, you could consider more idiomatic adverbs such as undoubtedly or confidently (as suggested by Dan).
Without doubt; certainly:
[Oxford Online Dictionary]
Contrasting "unworriedly expect amazing results" with "undoubtedly/confidently expect amazing results", the latter is more idiomatic. I agree with Dan that to anticipate is better than to expect.
You could say nonchalantly
(adj.) coolly unconcerned, indifferent, or unexcited; casual
"I shall prefer him for this task. And nonchalantly expect amazing result(s)."
You might use 'blithely':
adj. blither, blithest
1. Carefree and lighthearted.
[blithely. (n.d.) American Heritage® Dictionary of the English Language, Fifth Edition. (2011). Retrieved January 10 2016 from http://www.thefreedictionary.com/blithely .]
In your example,
I shall prefer him for this task, and blithely expect an amazing result.