The president wouldn't admit to have made a mistake.
admit to having made
Don't worry - I remembered to lock the door.
Correct
He denied stealing money from the company.
Correct (also: denied having stolen)
Kerry is used to getting her own way.
Correct
They tried to stop the fire, but without much success.
Correct
Her postcard made him thinking about his last holiday.
made him think about
He gave up making excuses and apologized.
Correct
He threatened telling the police if she didn't hand over the money.
threatened to tell
My friends encouraged me to stop smoking.
Correct
He was trying parking his car when the accident happened.
trying to park
I wanted him to call me a taxi and take me home.
Correct
Rachel regretted not to go on holiday with us.
regretted not going/having gone
My parents refused buying me a motorbike.
refused to buy
Luke suggested to order a takeaway.
suggested ordering (also: suggested that we [should] order)
Is it true that he's thinking to emigrate?
thinking of emigrating
Although he was exhausted, Mark carried on to run.
carried on running
Clare's mum didn't stop her to go out the night before her exam.
didn't stop her going out
Tanya's hairstyle was so strange that I couldn't help staring.
Correct (also: couldn't help but stare)
The prisoner begged allowed another phone call.
begged to be allowed
They dared him trying bungee jumping.
dared him to try