If I migrate the content from SharePoint 2010 to 2013 via third party migration tool, could we upgrade the site collection successfully passing by the custom solutions errors that might occur if database attach upgrade method is used?
If you migrate content only with the tool, yes you might bypass the warning messages you get when doing a database attach migration. You may still encounter problems depending on what these customizations did.
In our experience with using a third party tool to go from 2010 to SPO, all of the customizations we knew of were not carried over (obviously) and did not pose any major issues. There were a couple pages here and there using a custom web part. Those were stripped out when the pages were migrated.