It varies driver to driver. I think its up to the driver if they do...However Verizon requires a signature for all items. What I did was leave a letter and wrote my signature on it. He also gave me some Fed-ex slips that you can sign and leave on door. Maybe he was nice guy or something not sure. LOL
To be honest, they won't leave the package. They do this but in rare cases and only in case when the deliverable goods are not much expensive.
To my personal experience, i have never find this and they always take back the package.
Leaving the package at door and leaving it to get worst with sorrounding temprature or due to storm or rain is common problem with USPS.
I used to work in a company before which used to ship their products through USPS. There have been the same problem that the customer complain about the services from USPS. They leave the package at door steps and the things got worsen due to bad temprature or heavy rain.
The best way to avoid such situation, is to provide some other address for delivering. In case you have local depot you can have it delivered there to pick it up.
Or the best is to call their customer care and discuss the issue and they will provide you with the best solution..
Yes, Fed-Ex will leave the package. All that you have to do is leave a paper with the tracking number and your signature, and state on the paper that it is OK to leave the package. I've done it many times.