Tom skewed the survey results to back up his original claim.
"The car was parked skew (why not skewed here?)."
The first three are OK for me. The fourth doesn't work. We don't "lift something askew". "Askew/skewed" describe something stationary.
Crooked (adjective) and crookedly (adverb) are much more common. You should be able to recognize skew and askew, but you're likely to sound odd if you use them. Your example sentences are not very natural.
"Tim's car was parked skewed."
"He walked in a skewed manner."
"This dumbbell should be lifted up skewed not straight."
Please check.
What is the way to make them sound natural?
Tim's car was parked crookedly.
He walked crookedly.
The last one is more challenging because I don't understand what you have in mind. Maybe This dumbbell should be held at an angle when you lift it.