I certainly don't have sufficient medical/technical knowledge to explain exactly how the liver stores glucose - I don't know myself, but there again I don't need to know by what process it does it and hence have never bothered to delve into 'how' - so would suggest you look it up and come back and explain to us in simple terms.
It will continue to do it, as long as it gets signals from your body that other bodily cells don't have enough glucose - eg when our BG drops. However it does it in its own time. So - maybe I have a low because I've miscalculated how much insulin I needed to inject, to cover the food I was about to eat. I test after eating because I thought I felt hypo - and I was. I can't afford to wait for my liver to decide (because we can't know when it will do it) and I may have ;lost consciousness by then if I drop low enough - so I have to get my BG up quickly by eg drinking some Lucozade - which acts within 10-15 minutes and my BG has come up again. However - sometimes, because I was dropping very fast for whatever reason - my brain hasn't cancelled the message to my liver yet - so my liver will still 'dump' glucose into my bloodstream and raise my BG much higher than I wanted. I can't do anything to stop it - so I have to then give it a good while - several hours - to finally peak and stay there for a bit, until I correct the resultant high with a bit more insulin. If I correct with more insulin too soon - I'm in danger of this becoming a see-saw - hypo, then hyper, followed by another hypo and if I'm not careful it will get ridiculous. Up, down, up, down - never stable. So best plan is to wait for stability again - even if my BG is in the teens for a few hours - before I tackle the high.
The liver isn't able to multi-task - so if I drink too much alcohol too quickly, and then go hypo (cos alcohol lowers BG) - the liver will be so busy processing the alcohol (cos that's what it needs to do) it will NOT be able to help me out with glucose - so I could be not only tipsy/blind drunk - I'm also in SEVERE danger of going so very low, only hospital treatment will sort me out. It can be and is completely life threatening - so it's best just not to go there!