
Welcome to airbrushing!
Troublehsooting the blighters can be very stressful! The one where the paint comes out before you pull the needle back is suggesting a clog or something in the nozzle, along the needle, the needle seal etc, or even just ever so slightly impropper needle placement.
Firstly, whip the handle off the airbrush and loosen the needle chuck, and make sure the needle is sat fully flush in the nozzle and then re tighten the chuck.
If the problem persists, cleaning is required.
If the parts are infact clean as a whislte, perhaps lubrication is required.
I've used a bd130 before (well the same brush under another name) and the nozzle attaches via an awfully tiny fiddly thread with a little o-ring between the parts, take great care with this, It snaps off in the brush fairly easy (or splits the o-ring) when over tightened, and the o-ring makes too poor a seal if undertightened.
To lubricate the brush it wont take very much oil. There are airbrush specific ones, but any ultra fine oil should suffice. If in doubt shell out the few bucks on the specific one and it does last a long time.
I like to use skateboard bearing oil, mostly because its kicking around... Seems to work just fine, the tolerances in Skate bearings are very tight much like the airbrush so i figured its suitable.
It can interact with paint, so be sparing. You need it all along the needle, and in the trigger mechanism. My prefered way to coat the needle is to put a drop on a clean rag and then wipe the oily rag along the needle (being careful of the tip and not bending it!) - I tend to hold the needle at the back and pull it carefully through pinched fingers holding the rag.
The one where you rock the trigger back and forth could just be a build up of tip dry. Which is inevitable I'm afraid, especially with acrylics. Regular rinsing out can help reduce this, say for example you spray a few
ml of a colour, red? and then use it all up and need more, rinse the old red out before adding fresh ontop even though the colour remains the same, the older batch is slightly dryed and can cause clogs with lumps of dryed old paint sinking to the bottom of the new paint. It doesnt take a full on cleaning, I would shoot a drop of your chosen airbrush cleaner through to ensure theres not a build up of semi dry paint at the nozzle and then add my extra cup full.
It comes with experience with a particular paint, you will get to know how fast it drys up in the tip and how regular you need to flush out etc.
Best of luck, hope this helps

if your problem persists we might need more info to work it out.