When I release the clutch pedal, the clutch bites/engages immediately. There is hardly any travel and the pedal feels slack. Will I need a new clutch?

Sounds like the clutch cable needs adjustment so that the clutch is completely disengaged for shifting. Being a hydraulic clutch operation system there is no adjustment I'm afraid bar pedal height at the master cylinder push rod to pedal and that wouldn't solve the problem - as over all adjustment generally occurs as the clutch wears..so a good clutch would bite in the region of between the floor and mid way on the clutch pedal travel..if bite point is flat to floor then it possible you have drag on the clutch or its not clearing properly either a fault on the release system/hydraulic side either weak slave/master cylinder "prob try to bleed through again maybe and see if any improvement is felt"..or defective drive plate, buckled plate,weak pressure plate which releases the friction plate or seizure on spines/seized spigot bearing all of which would require inspection with the gearbox out. Hope this helps.

Your clutch disc is worn out.

Your pedal height may be adjustable by a stopper bolt at top of pedal. You would have to check the specification for your car. Most important that the pedal has about a half inch free travel before clutch begins to engage. You need that free play to ensure the release bearing (the throw-out bearing) is not touching the fingers on the pressure plate when not engaged. If it always is touching, it will be rolling around constantly and will go out prematurely.
Your new clutch will allow gear engagement sooner as the clutch pedal is released. I mean gears will engage just as pedal is lifting off from floor. You should notice that difference-the old clutch would have taken longer pedal travel from floor before releasing.

Engaging in the top third of pedal travel isn't unusual.

Clutch drag can be caused by a worn or broken diaphragm, damaged pressure plate, worn or damaged withdrawal mechanism or contamination of the driven plate, among other things...
There are two other causes of clutch drag worth considering - rusted or damaged first motion shaft splines or a bad clutch pilot bearing, if fitted.

In all cases the transmission will have to be removed for access.

The reason for the high biting point is simple. The clutch is made of 3 main components.
The pressure plate friction plate and the release bearing.
The more times clutch is depressed the concentric spring fingers wear in 2 ways . Firstly by good old metal to metal rubbing and secondly the concentric spring fingers don't return as far as a new one causing the release bearing or csc to adjust .
When the pressure plate concentric spring gingers wear they don't sit proud and require more pedal effort to depress. .
When they loose there spring like feature they require a greater distance from the release bearing. Or it has to travel further to depress them to lift The pressure plate and fee the friction plate from the lay shaft on which it is splined onto to give you drive once released.
The further back the bearing has to travel to allow the fingers to clamp the pressure plate to the friction plate . The more the clutch has become worn and unserviceable.
Self adjustment on hydraulic clutches are a hard one to gauge as you have no idea how far the piston has moved out to compensate for this wear . With a cable you can physically see the wear and adjust .
High biting points on any car are useless. You want a biting point to be 2 to 3 inches from the top and not nearly level with the brake pedal.
Anyway it seems apparent that Peugeot and Citroen both have high to almost no point at which they bite .
I myself would modify this as an engineer.
But it seems to be normal on these cats to have absolutely no biting point worth talking about.
My sisters 2012 206 1.4 is a very high biting point and I myself would not be happy with it . It gives u no indication of a worn clutch if its self adjusting.
Alll hydraulic clutches are self adjusting. So why are Peugeot and Citroens sooo high. Its ridiculous.
A very idiotic idea and I'm taking it back with her and I'm complaining. Hopefully I should get somewhere and ill update you all

once you have bled out all the air ,if the pedal still engages right down then suspect the fingers on the release plate are worn.Never replace a part of the clutch, allways change all 3 components,it save all the hastle of taking it to bits again and again.

Hard shifting may be caused be the clutch not releasing properly.If you have a new clutch your problem might be caused by your hydraulic clutch release system.When stopped and idling you should be able to shift gears without grinding.Reverse grinding may indicate a clutch release problem.You should have some clutch pedal travel up from the floor before you feel it engaging.The oil fill check is either at the speedo or a plug at the axle shaft level.Be sure to use factory recommended fluid.Gear oil is not a proper fluid.

this happens for a while untill everything settles,but the are adjustable slave cylinder rods,even after bleeding the system takes a while to work properly.

clutch needing replaced, very common on all manuals

pedal travel adjustment,yes
gear shift adjustment,no

you need to bleed the clutch system.. have someone pump the clutch pedal three or four times, hold it down and release the bleeder valve( make sure the person holds the pedal down till after you tighten the bleeder valve back down) repeat this four or five times or till the clutch pedal feels like it did before.. good luck keith...