I agree: Using “However” at the beginning of a sentence is fine. And the point made in #9 and #10 is important. (Anna, note how Stravinsky properly used the needed semicolon here: It was a great year for the company; however, the takeover bid was a mistake.)
We should emphasize, however, that however does not always require semicolons or commas. The OP’s second and third example-sentences correctly use “however” (with a comma before and after)--but those examples are correct only because the commas actually fit the sentence’s meaning and the intended meaning of “however.”
Look at the following definitions and usage examples from the (online) American Heritage Dictionary of the English Language.
In whatever manner or way: However he did it, it was very clever.
To whatever degree or extent: "have begun, however reluctantly, to acknowledge the legitimacy of some of the concerns" (Christopher Lasch).
In what way. Used as an intensive of how: However did you get here so soon?
In spite of that; nevertheless; yet: The book is expensive; however, it's worth it.
On the other hand; by contrast: The first part was easy; the second, however, took hours.
In whatever manner or way: Dress however you like.
Archaic Notwithstanding that; although.
Although some grammarians have insisted that however should not be used to begin a sentence, this rule has been ignored by a number of reputable writers. Forty-two percent of Usage Panelists say they do not follow the rule in their own writing, 19 percent say they observe it only sometimes, and 36 percent say they usually observe it.
For your reference, this advice from GrammarBook.com (“Commas”):
Use commas surrounding words such as therefore and however when they are used as interrupters.
I would, therefore, like a response.
I would be happy, however, to volunteer for the Red Cross.
And from GrammarBook.com (“Semicolons”):
It is preferable to use a semicolon before introductory words such as namely, however, therefore, that is, i.e., for example, e.g., or for instance when they introduce a complete sentence. It is also preferable to use a comma after the introductory word.
You will want to bring many backpacking items; for example, sleeping bags, pans, and warm clothing will make the trip better.
As we discussed, you will bring two items; i.e., a sleeping bag and a tent are not optional.
Use either a semicolon or a comma before introductory words such as namely, however, therefore, that is, i.e., for example, e.g., or for instance when they introduce a list following a complete sentence. Use a comma after the introductory word.
You will want to bring many backpacking items; for example, sleeping bags, pans, and warm clothing.
You will want to bring many backpacking items, for example, sleeping bags, pans, and warm clothing.
And from writingwithclarity.com:
What can be more difficult is determining when to use commas around them in mid sentence:
Yes: It’s a great idea. We don’t, however, have the budget. (However is used in contrast to what was said in the previous sentence.)
No: The deadline is Friday. We therefore need to work overtime to finish. (Therefore is essential to the main thought of the sentence, so it is not set off with commas.)
Use your voice as a guide. Read the sentence aloud. If your voice rises as it pronounces the adverb, that tells you the word is essential to the main thought and therefore not marked off by commas.
[End of quote.]