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Yes a carbon stem should work fine with aluminium bars, providing they have the same fit size, usually 31.8mm. Similarly carbon bars are fine with an aluminium stem. Take extra care to adhere to manufacturers torque settings when fitting, use a torque tool if you are unsure.

Providing the clamp size is the same as the handlebar diameter, technically MTB and road bars and stems can be switched. However! It is worth bearing in mind that the manufacturers intended use for a road stem is on the road, and likewise mountain bike components are designed for use on mountain bikes. Mountain bike parts are generally made stronger than their road bike equivalents, so it would be advisable not to use a road bike stem on an enduro mountain bike.

Yes, you can mix and match different bar & stem manufacturers, providing the have the same handlebar diameter and stem opening. Looks nicer when they match though!

The easiest way to learn to wrap bar tape is to watch a Youtube video, Park Tools and GCN have easy to follow instructional videos covering pretty much everything to do with looking after bikes.

Thicker bar tape (such as Lizard Skin DSP) is a great way to reduce vibration fatigue in hands and wrists. Alteratively, gel strips (stuck to bars under regular tape) can also help with the same problem. Gloves with extra padding can also help.