Could you, for example, have one of your craft skills filled in as carpenter, but have no ranks in it? Being a very bad carpenter, but a carpenter nevertheless.
Not only can you, you have to. You never roll a “Craft check” or “Perform check,” those skills don’t exist. Rather, they are categories of skills, like “Craft (basketweaving)” or “Perform (underwater basketweaving).” That’s still true even if you have no ranks, and it’s relevant for things like feats or instruments, which may need to be compatible with the chosen Craft or Perform skill.
Craft and Perform may be used without ranks, so if you have high Intelligence or high Charisma, and/or other bonuses to those skills, you could even be good at them without any training (e.g. ranks). It’d be hard to keep up with someone who is actually training it, though.
Profession’s much the same, but it actually goes beyond that. As a Trained-Only skill, you cannot beat any DC higher than 10, no matter what you roll, if you have no ranks in that specific Profession skill. So having ranks in Profession (woven-basket critic) doesn’t let you succeed on DCs greater than 10 on Profession (woven-basket recycler); they are separate skills, and you have no ranks in the latter.
You can Craft and Perform, but you can't use Profession;
As stated here, Craft and Perform can be used without training. This means that you don't need to have ranks to use any of the Craft Skills (not limited to Alchemy, Armors, Bows, Traps, Weapons, Varies) or Perform Skills (not limited to Act, Comedy, Dance, Keyboard Instruments, Oratory, Percussion Instruments, String Instruments, Wind Instruments, Sing).
Unfortunately, this does not apply for Profession, and you'll have to get at least 1 rank in a Profession to use it.
It's worth noticing that every failure in a Perform check will give you a -2 when trying to Perform in front of the same audience (because they were not impressed by you and now have prejudices). Be aware of that and use Perform wisely when you don't have ranks in it.
If you don't want to get ranks in these skills, there is still something we can do to improve your checks.
Artisan's Tools: You may want to get these to get a +2 on every Craft check;
Gnome Racial Trait: Gnomes get +2 to a Crafting Skill of their choice.
Masterwork Musical Instrument: If used during the performance, it will grant a +2 on the Perform check.
Skill Focus (feat): Choose a Skill; Without any rank in it, this feat gives you +3 on every check for that Skill (you can get up to +6 when you have 10 ranks, but this is not the case).
Yes: from a roleplaying point of view, you can say that your character is a carpenter by trade, but one who never actually learnt any carpentry as such and just got by on intelligence (literally).
As far as I know, there is nothing special about "filling in" a Craft skill on your character sheet; sheets just have several Craft slots in the skill list so that you can fill them in if/when you take ranks. There is no reason I know of why you can't fill one of these in without actually taking ranks, and several reasons why you might want to: for the above roleplaying reason, or because you're a gnome and need somewhere to put your racial bonus.
Yes, and you don't even have to be a bad carpenter, except that everybody who is not a good carpenter, is a bad carpenter. All skills you haven't put any points in, exist as untrained skills. And if it's a skill you can use untrained, you can use it. All sub-skills of Craft and Perform exist, you have access to them, so feel free to write them on your character sheet if you expect you may use them. Though rather than Craft (carpenty), it might be better to write down Craft (untrained), because the same number will apply to all your untrained Craft skills. Unless you get specific bonuses for one, or you want to write one down now because you expect to put ranks in it later.
If you get any bonuses to skills, you get them also to untrained skills. If you can use the skill untrained, you can use that bonus. If you can't use the skill untrained, you can't make use of the bonus (but you can as soon as you put a rank into it).
Anyone can try to make a table, weave a basket or use any other craft skill. And with enough talent (high ability modifier, or misc bonuses), it might even be a good one, despite the lack of training. Same goes for Perform. Profession skills can't be used untrained, however, so if you list it as a skill, you might forget that you can't use it, and accidentally use it untrained.
Using untrained skills is also relevant for Bards. The description of Bardic Performance suggests they need to use a Perform skill, but they usually don't have to make a skill roll, and it doesn't matter which Perform skill they use. So a Bard whose best skill is Perform (strings) can choose to use his untrained Perform (oratory) or Perform (comedy) for his Inspire Courage.
Knowledge skills also have a lot of subskills, and while they normally can't be used untrained, a Bard can. And considering how useful knowledge skills can be, it can be a good idea to write Knowledge (untrained) below all your trained Knowledge skills. In fact, PCGen does that for you automatically.