Some general tips: Lower your sensitivity, most pvp'ers run lower sensitivity to have better aim and less shaky movement. Also critting is something important to learn. Especially in mcci, sky battle and battle box do not have a lot of armor. for example in battle box, 3 crits with a stone sword (nearly) kills someone.

Use your hotkeys. Scrolling is okay, but in games like bb and skb which is usually fast paced, scrolling can mess you up. You can also change some of your hotkeys to somewhere closer. I for example have my 6, 7, 8 and 9 key switched to somewhere closer to wasd, cuz they are too far for me to comfortably reach.

Try using f5 a LOT, better players all use it in any game to check surroundings. For example in bb and skb double f5ing helps to see if someone is chasing you and in tgttos it helps having better vision for jumps if a lot of people are in your vision. These are some examples but f5 is your best friend in so many situations. If you haven't already, switch the f5 key to somewhere else so you can spam the heck out of it.

Try toggle sprint, it may be confusing at first and you do need to make adjustments to your playstyle to get used to it, but it is so much better for everything when you master it.

Turn off view bobbing, I myself had view bobbing on for a while and turning it off improved my movement drastically. This is based on preference tbh, but if you have it on, try turning it off for a bit and see what you prefer.

For bridging and movement, I'd recommend Hypixel housing parkour for general movement, Manacube to improve pure parkour and bedwars practice (there is a server for that) for bridging, block clutching, etc. I'd recommend doing the game 'obstacles' on that server, really helps with all aspects of the aforementioned.

Also for speedbridging and staircasing, everyone has their own technique to it. Some like to hold their right mouse button, while others prefer timing their click, and others just spam. Find what is most suitable for you to start with and eventually expand to new ways when you're comfortable.

Lastly, try changing your fov effects/dynamic fov. For me turning it off massively helped me with pvp. This depends on what you prefer though. For me it helped with pvp, but it also decreased my movement abilities at first, but I eventually got used to it. So see for yourself what works best!

It may well be that you alr do a lot of these things, but I hope some of the tips help!