Sure, let's go through each sentence and choose the correct item, providing explanations for each choice:
He avoids travelling by plane because of his phobia.
- The verb "avoid" is followed by a gerund (the -ing form of a verb). Therefore, the correct choice is "travelling."
Helen will never forget dancing with Tom Hanks at the party.
- "Forget" can be followed by either a gerund or an infinitive, but the meaning changes. "Forget doing something" refers to forgetting a past event, while "forget to do something" refers to forgetting to perform a future action. In this context, "dancing" is a past event, so "dancing" is correct.
Let’s spend our weekend shopping downtown.
- The verb "spend" when referring to time or money is followed by a gerund. Thus, "shopping" is the correct choice.
Mark has decided to become a lawyer like his father.
- The verb "decide" is followed by an infinitive. So, "to become" is the correct choice. Also, it's more common to use "like" rather than "as" when making comparisons or showing similarity, so the phrase should be "like his father."
If you want to be successful you should use your imagination.
- The modal verb "should" is followed by the base form of the verb, so "use" is correct.
Julia is looking forward to getting a mail from her boyfriend.
- The phrase "look forward to" is always followed by a gerund. Therefore, "getting" is correct.
Looking at Tom’s funny face I couldn’t help laughing.
- "Can't help" is followed by a gerund. Hence, "laughing" is correct.
Dad, really I promise to look after my younger sister.
- The verb "promise" is followed by an infinitive. Thus, "to look" is correct.
What you can do is just not to mind your business.
- This sentence is a bit awkward. Typically, "mind your own business" is the correct phrase, and "What you can do" should be followed by the base form of the verb, so "do." The sentence might be intended as irony or sarcasm.
Don’t let him eat ice cream or he will get a sore throat.
- The verb "let" is followed by the base form of another verb, so "eat" is correct.