I see that some Professions have skills bound to F1, F2 and so on. Yet some have 0 skills bound there. Are these Profession specific? And if so do you have to unlock them or spend points on them or do they become available naturally?
Each class has a specific "F" skill, or multiple "F" skills. Some only have a single ability on F1, while others have 3 or 4 going up through F2, F3 and F4. These are unlocked at different levels, depending on the class, and in some cases depending on your current weapon/skill loadout (or pets when it comes to the ranger).
Warrior - F1 only. Burst skills. Dependent on the weapon you're currently equipped with. A bar fills up with "Adrenaline" as you do damage to your opponents, and the more you have stored up, the more powerful the Burst skill becomes.
Elementalist - Attunements. Fire (F1), Water (F2), Air (F3), Earth (F4). These skills offer the Elementalist the ability to determine what their 1-5 skills will be with their current weapon set up, offering the Elementalist up to 20 available skills per weapon set. For this reason, the Elementalist is one of two professions that does not get a weapon swap.
Guardian - Virtues. The guardian has 3 Virtues (F1, F2, F3). These are self only "buffs" that offer unique attributes to the Guardian. When used, they offer their bonus immediately to all allies in the area at the cost of going on cooldown and not offering their passive bonus to the Guardian until they are refreshed.
Necromancer - F1 only. Death Shroud. Puts the Necromancer into a ghastly form, giving them a new bar that counts as their health, as well as a new ability bar. When the health bar in Death Shroud is depleted, the Necromancer merely reverts back to their normal form, rather than dying.
Mesmer - Shatter Illusions. Each of the 4 shatters provides a different effect, and is affected by the number of Illusions / Phantasms currently active. Your shatters can explode for damage, apply stacks of confusion, apply dazes, or provide a unique buff to the Mesmer called "Distortion" which is, essentially, a short invulnerability.
Engineer - Tool Belt. Based on the Engineer's healing (6) slot, as well as utility (7, 8, 9) slots, the Tool Belt is usually a one-off effect of the respective slot. For example, having the Mine Kit equipped in your 6th slot offers a unique mine-based ability on your F2 key. Because the Engineer has such a versatile loadout with utility / healing weapon kits, as well as F keys being influenced by loadout, they are one of the two professions that do not have a weapon swap function.
Ranger - Pet abilities. Each pet has an active ability that is controlled by the F keys, as well as the ability to command your pets and swap pets.
Thief - F1 only. Steal. Thieves can steal from (supposedly) any hostile NPC or hostile PC, providing them with a pseudo-random ability in return. This ability that they gain is a 1 time use.
Every profession has a unique mechanic which makes use of the F-Keys. For example, the Necromancer has the Death Shroud which is activated with the F1 key, and the Ranger controls their pet by using the F-Keys.
The F-Key mechanics unlock as you level up; you don't need to spend any trait or skill points to unlock them.
The F-Key skills are specific to each profession, and they behave like any other skill. They have a cool down, and there is no "skill tree" associated with them that you need to invest points into. It is possible to improve the F-Key through certain Traits though, but this varies from profession to profession.