Offers, as stated above, are promises (usually). Note that Jane's offer to Mary over the phone was a PROMISE - 'I WILL pay you $6000 if you WILL sell me your car'. The hypo even uses the word offer! This is without question a Bi K - it is certain that Jane is bargaining for the promise to sell the car. What would be the performance - handing over the keys?
To stray slightly from the issue, the simple fact that something identifies itself as an "offer" is not dispositive that it is, in fact, an offer. The offer is that which creates a power of acceptance in the offeree. Ads generally fail as offers because they usually do not specify who has the power to accept, they often do not specify a value of the offered goods, and they almost always fail to manifest an intention to be legally bound. A famous example of an ad that IS an offer is Carlill v. Carbolic Smoke Ball Co. 1 Q.B. 256 (1893).
Granted, I agree with you that the offer is not the newspaper ad but Jane's initial telephone call; but hey, we're in law school, and we live to argue.
I agree that the promise to pay $6000 was consideration for Mary's promise to sell the car to Jane. A promise can indeed be consideration--and fortunately the Restatement (Second) of Contracts § 75 (1981) provides under what circumstances a promise can be consideration: "...A Promise which is bargained for is consideration if, but only if, the promised performance would be consideration."
The promised performance is the payment of $6000. This would be consideration. Therefore, as you have said already, the promise to pay $6000 for the car is also consideration.
When analyzing bilateral vs. unilateral contracts, rather than looking at promise or performance as acceptance, I prefer to defer to Murray's analytical tool: At the moment of the formation of the K, if one party has a right (but no duty), and the other party has a duty (but no right), then there is a unilateral K. If at the moment of formation of the K both parties have both a right and a duty, then it is a bilateral K.
In this case, both parties have rights and duties; Jane has a right to the car and a duty make good on the promise to pay, and Mary has a duty to transfer title and possession of the car to Jane and a right to Jane's $6000. Therefore, it is a bilateral K.