Right so there's no "optimal" method for this, brewing is a balancing act between sweetness, alcohol, carbonation-pressure and yeast. Even things like the shape of the fermentation vessel effect the performance of the yeast. You will always need a little trial and tweaking, and this is part of the fun.
So a ginger beer "bug" (or "plant") is typically a wild yeast culture in a gingery sugar-syrup. Some recipes call for the addition of brewer's or baker's yeast into the mix. For me, doing this ceases to make it a bug/plant. If you want to use cultured yeast, that's fine (and probably a good idea for repeatability), but it's no longer a bug/plant.
How can you make a bug? Generally you start with a sugar water of a ratio 1:10 by weight. That's 10° Plato, or 1.040 specific gravity. Boil this to sanitise it. When we made these as kids, we always added a teaspoon of ground ginger too. I don't know for certain what this brings to the party, ginger is anti-bacterial though. When it's cool we added some sultanas (dried grapes). The yeasts on the skin of the fruit will colonise the sugar-water. You could probably use any sort of fruit-skin here. Honey naturally also contains a lot of wild yeasts (as long as it hasn't been pasteurised etc.). Cover your bug with plastic-film, or another sort of covering. Do not screw on a tight-fitting lid! The yeast will produce gas and pressurise the container.
Since we're talking "optimal" methods, it would be best to add some yeast nutrient to this sugar-water, since pure sugar is a bit unhealthy for yeast (follow the manufacturer's recommendation on amounts). Or use malt-extract instead of sugar, since it contains more of the minerals yeast need to grow well.
Sugar-water is also a great environment for bacteria. Pre-acidifying the sugar-water to below 4.5pH with some kind of food-grade acid, like citric acid, will reduce the chance of bacterial spoilage. Alternatively the sugar-water could be boiled with the addition of a few hop-pellets - the hops don't need to be boiled for their anti-bacterial properties to work, it's just a convenient time to add them.
Leave the bug (with fruit) for a few days in a warm (~22°C / 70°F) area. If it's too cold the yeast will not grow as quickly. If it's too hot the water will evaporate. In a few days to a week you should see some bubbles and a scum forming on the top of the liquid. This is the yeast growing. There may also be some other moulds (yeast is a mould). If these are brightly coloured (red/yellow/black), it's not safe to eat, discard the bug. After a week (or two) the yeast will consume all the sugar in the sample, and drop to the bottom. It will form a creamy-white film on the bottom of the jar. You can use this amount of yeast right now, or decant the liquid off the top and add more sugar-water to grow the yeast further. It depends on the batch-size of your ginger beer as to how much yeast is needed. It will still work no matter what, but the time taken will be longer with less yeast.
But! Do you even care that it's made with wild yeast? If not, it's possible to simply use a packet of brewer's yeast, or even bread yeast. This will be more reliable, simpler and quicker. Personally, I would use a teaspoon of bread yeast, or brewer's yeast to be fancy (e.g.: Safale US-05 ). But I always have bread yeast on hand.
So we've just about covered the yeast / bug / plant side of things. There's plenty of further reading on this. Essentially you're "Capturing Wild Yeast". Duck-Duck-Go that term.
Right so that covers the yeast. Now onto the beer...
First a simple fact about fermentation: Unless the amount of alcohol has become toxic to the yeast (typically 12~14%), or until the sugar is all consumed, fermentation will continue. To stop fermentation you must pasteurise the beer. Putting the beer in the fridge will only slow it down. It's just about impossible to filter out the yeast on a home-brew scale. Even clear beer can have ~100 million cells / ml. Additives like potassium sorbate can be added to stop the yeast multiplying, but that's getting into quite an advanced topic.
So ... because the yeast keeps processing sugar, people are naturally quite wary about putting yeast-carbonated beverages into bottles. The pressure builds up to the point where bottles can explode. Even champagne bottles (typically rated to 7 volumes of CO2) can explode. A PET plastic bottle (assuming it's un-scratched, etc.) can hold a little more pressure than a normal glass soft-drink/soda bottle - but when it breaks, there's no flying shards of glass. Quality used (or new) PET bottles would be a good choice for ginger beer storage. Do not use bottles for pressurised drinks unless they are designed for it. Cheap glass bottles from homewares stores typically are not OK. Use recycled PET or beer bottles instead.
When a yeasty beverage is put into bottles for carbonation, this is under the assumption that the existing fermentation is complete. A small amount of sugar (typically 5 grams per litre for ~ 2 volumes of CO2) is added so the yeast re-starts fermentation to carbonate in the bottle, but without going over the pressure rating of the vessel. So with your flip-top bottles, do not seal them until you are sure fermentation is complete. How do you know? Use a hydrometer / saccharometer, or taste it. If it's still sweet, it's (probably) not done.
The more sugar you put into your beer, the more alcohol it will eventually contain. About a 10% sugar-water (by weight) will typically ferment out to 4-5% AbV alcohol. Strongly sugared environments become difficult for the yeast to ferment, so if you add a lot of sugar, the yeast will ferment to the point where the alcohol is toxic (to the yeast), but still leave some sugar behind. However, this makes the beer difficult to then carbonate. The large concentration of sugar is itself bad for the yeast because it puts osmotic pressure on the cell walls, so this can also result in a poor fermentation.
During fermentation the yeast will foam up, putting a ring of gunk around the top of the container. This is absolutely normal. I would allow at least two weeks for fermentation to finish at room temperatures. If it's too cold (say < 17C) fermentation will slow dramatically, but will re-start when temperatures warm. It's best-practice to use a hydrometer to check the progress of the fermentation, but many people do without them.
So say you started with a 10% sugar (by weight) water, with some amount of ginger and (maybe) lemon to taste. You fermented with enough yeast, at room temperatures. After about two weeks, it would be normal to expect fermentation to be complete. Your hydrometer reading would be around 1.005 (very low), it may even be less than 1.00. Dissolve 5 grams per litre of table-sugar to the beer, and put into santised pressure-rated bottles. In another week (or two), these should be carbonated. If you use PET bottles, simply squeezing the bottles will allow you to determine if carbonation has happened.
If this beer is too dry (that is, has little remaining sweetness), back-sweeten in the glass with some sugar, honey, etc. You can of course also drink the beer mid-carbonation, putting it into the fridge to slow (but not stop) fermentation; then drink them when desired.