Since you are doing extract, are you doing partial or full boils boils? Partial boils are easier to cool with an ice bath in the sink or other container like you are doing.
Depending on how bad you burned your LME, the batch may have still turned out OK. Never toss a batch unless you absolutely have to. In the future, continue to turn off the heat or just add the LME slowly to avoid the scorching.
Kegging is faster than bottling, but I suggest not expecting the beer to be ready in two days. It is still green at this point and will benefit from aging. You will be IMHO much happier building a pipeline of beer so that you always have some ready rather than rushing it from fermenter to glass. Typically for my beers I let the keg sit and age for three weeks before carbonating. Then inside my kegerator I have two kegs on tap and one connected to co2 at normal serving pressure. At normal serving pressure and temperature the beer will be perfectly carbonated in a week to ten days. That two days at high pressure or shaking business often over carbonated my beers and just didn't turn out as well as being patient.
Hops are quite easy to grow if you have the space for them. They grow quite tall and need a structure that allows them to reach often in excesss of twenty feet or you can train them along a garage or fence top. Expect a couple of years for them to produce enough cones to brew with.
BTW, brewing beer is the best hobby I've ever had. No regrets
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