Being incorrect might make you happy. While some style rules prohibit it, it is completely fine to begin a phrase with "but" while writing. By all means, begin sentences with "but," but keep in mind that "but" also belongs after a comma.
If you want to give a specific example of when it is acceptable to start a sentence with "but," here are a few cases where it works:
When giving an exception to a rule: "Eating meat on Friday is not allowed for Jewish people, but Hal Wolfe from Toronto will be eating meat on Friday night."
When discussing two things simultaneously: "Staying in today because it's Tuesday and staying in tonight because it's Thursday."
When giving a partial explanation: "I didn't go to college because I finished high school early, but I would like to become a writer."
When asking a question without answering it yourself: "Do vegetables help you lose weight? But isn't it better for your health?"
When commenting on one aspect of a subject: "Calling out students for chewing gum during class is okay, but telling them not to chew it at all is over-the-top."
Using any style peculiarity excessively ruins your work.
Yes. Although, starting a phrase with "however" is absolutely appropriate. Indeed, beginning a statement with "although" should be encouraged rather than discouraged. Using this word at the beginning of a sentence is known as an oxymoron.
The word "however" is used to show that something else could happen but that it would be unusual or unexpected. For example, you might use "however" to say that someone who usually works very hard doesn't have to work so hard today because it's a Saturday. Or you could say that although John loves his job, he'll still need to find another one because his current one has been sold.
Here are other examples of using "however":
"However," said Alice, "I don't think I shall go down my own rabbit hole."
"However," said Bob, "I do believe I can fly under the rope bridge."
"However," said Susan, "I will not put my hand into the fire."
Or: "How ever," replied the fox.
Yes, the answer is yes. Starting sentences with the conjunctions "and" or "but" is totally acceptable. It is, nevertheless, a little casual. If you want to be more formal, use a more formal language.
We frequently create run-on sentences because we believe the words are connected and that separating them with a period doesn't seem acceptable. For example, if we wrote, "She enjoyed skiing, but he didn't," we'd have a comma splice. However, since periods are used to separate sentences, leaving out the punctuation makes our sentence sound wrong.
The most common cause of the run-on sentence is using too many subjects or verbs. If you do this, sometimes it can be fixed by adding a few more objects or switching some of the subjects or verbs around. For example, instead of saying, "I like my job, my coworkers, and my boss," say, "I like my job. My coworker is helpful and my boss is understanding."
Another cause for the run-on sentence is using phrases rather than complete thoughts. For example, instead of saying, "I like eating ice cream every day at noon," say, "Eating ice cream at noon is one of my favorite things about living in Arizona."
Finally, a run-on sentence may also result when there is a lack of coordination between the subject and verb. For example, instead of saying, "I like skydiving, swimming, and camping," say, "Skiing is my favorite sport because it's exciting and challenging. I also enjoy swimming and camping."
Another reason people believe you can't start sentences with coordinating conjunctions is that it converts the phrase into a fragment. This misunderstanding may stem from a misunderstanding of what conjunctions are. Coordinating conjunctions connect two items in order to show relationship or contrast between them. They have no independent meaning themselves; they're just connectors.
Starting sentences with conjunctions is acceptable in modern English. It is actually quite common among writers for various reasons. The most common one is that it provides clarity for the reader. If a writer wants to emphasize a particular idea within the sentence, they will often do so by starting the sentence with a conjunction.
It's totally OK to begin a statement with "I," "but," "and," or whatever. It is absolutely acceptable for a short essay to include four or eight paragraphs rather than five. The passive form is perfectly acceptable. So are gerunds and present participles.