Can a character without a skill use a card that adds to that skill?
After reading some other forums, I believe the answer is actually Yes
Note: someone of the below is copied from David C Ellis's answer.
From the Paizo FAQ's Attempting a Check section:

Cards that require a check specify the skill or skills you can use to attempt the check. Each check to defeat or acquire a card lists one or more skills; you may choose any of the listed skills for your check. For example, if a check lists Dexterity, Disable, Strength, and Melee, you may use any one of those skills to attempt your check.

and

Even if your character doesn’t have any of the skills listed for a check, you can still attempt the check (unless you’re trying to recharge a card; see Recharge on page 15), but your die is a d4.
(Recharge on page 15 reads:)if you don't have at least one of the skills listed for the check, you cannot attempt to recharge the card

and

Players may not play cards that modify a skill unless you’re using that skill, and players may not play cards that affect combat unless you’re attempting a combat check.

So based on the first and second quote, you have to choose one of the skills listed on the card you're attempting to acquire/defeat/etc. even if you don't have that skill on your character card, but your die will be a d4. The wording &quot;Even if your character doesn’t have any of the skills&quot; simply suggests an extreme case, it doesn't mean that's the only situation where you can use a d4. It implies that you may use a d4 for a skill check with a skill you don't have, even if there are other skills (that you do have) to choose from. The only exception is recharging. In that case, you must recharge using one of the skills that you do have.
Then, based on the third quote, you can only play cards that modify the skill you chose so if you chose melee (even though it's not on your character card), you may use an ally that modifies your melee check. This third quote just states that you may not use an ally that, say, adds a die to your intelligence check if you chose melee as your check attempt skill.