As Kenny has said in his comment (above), it's very common to use 'me' in this way. In fact, Kenny has also used 'me' after 'showed' in his corrected version of your second sentence. The way you have used 'me' in this structure is perfectly correct.
With verbs of action, like 'walk', 'show' and 'ask' we normally use the past form as you have done in your second example.
However, with stative verbs, such as 'want' we do use the present tense because those states or conditions continue over a period of time.
So the structure is:
Noun/pronoun + bare infinitive or past form if verb of action + noun/pronoun + infinitive
The suggestion that you say "He asks me to cook a dinner" is a grammatically correct option in English, but would be unusual and only occurs in some forms of story-writing.
We can use the present tense to express a habit and for example say "He asks me to cook dinner" + "every week/twice a year/often/rarely/etc" - but note that this sentence doesn't include the article "a" because there is more than one dinner.