Moreover and furthermore are essentially interchangeable synonyms. They're formal substitutes for additionally, also, as well, in addition to, likewise, and too. Those two words are common in academic prose, partially because writers don't combine sentences well or often enough, partly because they need to use them to keep their sentences short enough to read without causing reader fatigue, and partly because it's the easiest way of adding additional information. I try to eliminate as many instances of moreover and furthermore as possible, but it's sometimes impossible.
In addition, they emphasize the "too-ness" of what comes next: {Moreover/Furthermore}, Judge Jones owns stock in the defendant's company, so he should recuse himself from presiding over this case.
The one you use is pretty much a question of personal taste, but my rule of thumb is to put moreover first and furthermore second if the two appear in the same paragraph on in contiguous paragraphs. Others may have a different "rule", but I think it's strictly personal choice.
Addressing your specific questions:
'Can you imagine how dirty it gets over time?'
When people talk informally, they are often a little sloppy. The present tense would be the perfect choice to state a generalization or natural law such as:
'White gets dirty over time.'
This idea was close enough for the speaker. Someone who likes to be more precise might say, instead, "Can you imagine how dirty it would get over time [if we bought the white pillow]?" But you might not enjoy living with someone who uses English precisely all the time. Such people tend to be nitpicky.
'The grease from your fingers will burn into the bulb and then it breaks.'
Here, two ideas have been spliced together into one sentence.
First idea: "[If you touch the bulb with your fingers] the grease from your fingers will burn into the bulb."
Second idea: "When that happens, i.e. when you touch the bulb with your fingers, the bulb breaks."
This is another example of the simple present being used for a generalization or natural law. And again, in a situation of informal speech you caught the speaker being a little sloppy.
'I hope it goes away overnight.'
I don't know how the English grammar experts would view this (if you want to know, you could ask over at ELU SE), but I will share how I see this example. My other primary language is Spanish, which has a subjunctive. English has one too, but people don't think about it much. In Spanish you really can't get away without thinking about it. In Spanish, this would be
Espero que se quite para mañana. ("se quite" is conjugated in the subjunctive)
I see the English sentence the same way -- I see goes away as the present subjunctive, which happens to be conjugated the same as the simple present.
Your questions were good, and you've done some careful listening and recording of what you've heard.
I would be remiss if I didn't point out a small but important misuse of the simple present tense in your question:
You wrote: I moved from Germany to California and since I'm here I hear people use the simple present....
Expressions beginning with "since" are notoriously easy to get tripped up on.
Better: I moved from Germany to California and since I've been here I've heard people use the simple present....
Or: I'm from Germany. Here in California I hear people use the simple present....
Best Answer
Downside is more modern/informal, and a bit slangy (some would still call it "business jargon").
Interestingly, people sometimes head up the section on, say, a business case assessment, with singular Downside - followed by perhaps several separate disadvantages. They usually head it up with plural Downsides though. But it would be very unusual to use Drawback or Disadvantage in the singular there.
Drawbacks is somewhat more informal than disadvantages, but they're both fine for almost all contexts.