Not necessarily. It is better if a contract is evidenced in writing but contracts can just be a verbal agreement. Contracts can also be evidenced by action. For example, you made an agreement for someone to do work on your behalf and you transfer money into their account as payment.
Yes. The sale of land has been held to be a valid agreement even though there was no written contract. The parties agreed to the terms of the deal in an email. One party then tried to renege on the deal and it was held a valid contract of sale had been formed.
A deed is a contract that does not have consideration passing between the parties. For instance, in a contract a sum of money or some other form of payment is transferred. With a deed no money or other payment may ever change hands. For instance a deed could be signed which prevents one party from doing an act that they would otherwise be able to do.
The execution of a contract and a deed may have different requirements. Deeds are usually witnessed by a party that is not involved in the transactions. With both deeds and contracts it is important to ensure the document is signed correctly. This is especially so if one of the parties is a corporation.