Some context: this was 130 years ago, back when the US had an okay
but certainly not top-tier
navy, and a relatively-weak army. We've got some hindsight to see how things played out.
On annexations of islands:
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The Cook Islands today are a country in free association with New Zealand.
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Fiji is an independent country.
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Hawaii was annexed in 1898 and became a US state in 1959.
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I'm not sure why Hawaii and the Sandwich Islands are listed separately. Might be terminology in 1895 differed from present-day terminology.
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Cuba is an independent country.
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Haiti is an independent country.
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I think "Friendly" refers to Tonga, which is an independent country today.
Tonga became known in the West as the "Friendly Islands" because of the congenial reception accorded to Captain James Cook on his first visit in 1773
On "Licking John Bull out of his boots"
at the time, the British Empire and the US considered each other fairly likely candidates for fighting in a war, made war plans for each other, and the conflict never actually happened. After the US wound up fighting alongside rather than against the British in World War I and World War II, the two wound up allied.
On "sweeping his enemies from the seas", yes; the US is the biggest naval power (and allied with most of the other substantial naval powers). We'll see where the growing China rivalry goes over time, though; in 2024, China has more warships than the US, though the aggregate tonnage of US warships is significantly larger than China's.
On "establishing formidable and invulnerable coastal defenses": well, not really in the sense that Puck would have thought of it, with naval forts and guns, but due to aircraft and warships, it could ward off a naval invasion easily, so kind of functionally similar.
On the Monroe Doctrine: I'm not sure that it's quite as meaningful today; it really dealt with an era where there was a potential "scramble for the Americas", where the US didn't want opposing major powers entering the Americas. Kerry called it obsolete, Trump's referenced it. I suppose if a major power started annexing new chunks of the Americas, the US would probably take issue with it today, but unless China decides to do so, I don't think that anyone's likely aiming to do so, so...shrugs
EDIT: Oh, and one other note: the "torpedo" in the image will refer to something akin to what we today would call a "naval mine". Terminology shifted.