Is there a way to forward email to another email account but still to keep the email in the original inbox?
This does not appear possible according to Verizon's instructions about email forwarding.
If you are trying to forward all emails as they come in the answer is no you can not have them in both places. "To forward the email individually open the email you wish to forward and click on Actions and select forward.
If you set your verizon.net email to forward, it will NOT keep a copy of the message on the verizon.net site. It is possible to set this up with your verizon.net address, but it requires the use of filters.
After some trial and error, this is the way that I've found to make this work with a verizon.net address successfully. You have to set up 2 filters.
If To or CC matches email@example.com then automatically move to folder: Inbox
This seems redundant, but this is how you get it to keep a copy of the message in your verizon.net inbox. Replace "yourusername" with your actual verizon.net username.
If To or CC matches firstname.lastname@example.org then automatically send notification to email@example.com
This sends the message to your other email account. Replace "firstname.lastname@example.org" with your 3rd party email address.
BOTH of these filters have to be set up for you to get a copy both places.
You can also change the criteria if you don't want every message to forward. Hope this helps!