Pardons are a specific type of executive order, which are a legal document. There are expectations of legal documents, but this is dealt with in common law, not in the constitution.
As a matter of common law, a definitive statement on whether you can pardon by tweet (for example) could only come from a court, and so would only occur if there was dispute. If a writ of habeas corpus were issued, and the prisoner produced the napkin claiming it to be a presidential pardon, then the court would have to inspect it. In principle, common law allows for most legal documents to have any form, but proving that a document is genuine and valid when it is not in the conventional form can be difficult.
So in terms of the specific questions: a napkin, tweet or verbal order may be sufficient to establish that a pardon has been granted, but would almost certainly come under close scrutiny, especially if the president denies the pardon at a later date. If the President intends the pardon to be valid, it is in their own interest to have it properly typed up, signed, dated and published.
A "get out of jail card" can't be issued because a pardon must occur after the commission of the crime. It is possible to inform someone that you would pardon them, and it is possible to pardon someone prior to inditement, but the president can't publically or privately pardon someone before they have committed a crime. If a president wants to get an aide to break the law for "patriotic" reasons, what they usually do is issue the pardon on the 19th of January in their final year. To make the pardon easy to enforce it is signed, and published.
In principle neither a signature nor a date is required, in practice it is easier for a President to get the pardon accepted if it is signed.
In many ways this is the same check-list as for other legal documents: A contract can be on a napkin, in a tweet, or agreed verbally. But if you want your contract to stand up to inspection in court you'd better get it notarised, signed, countersigned...