Classic Outlook (Windows)
File, Options, Mail, Signatures. Create new, paste the signature from Outromail, set it for new messages and replies. Classic Outlook renders with Word's engine; Outromail's tables are built for it.
New Outlook and Outlook on the web
Settings (gear), Accounts, Signatures. Paste, name, choose defaults, save.
Outlook for Mac
Outlook, Settings, Signatures. Click +, paste, close. Then set defaults under “Choose default signature”.
Rounded corners
Classic Outlook ignores border-radius, so the card shows square corners there. Everything else renders the same.
Images as attachments
That happens when the signature contains embedded images. Outromail uses hosted images by URL, which Outlook displays inline.