Writing : I'm finicky about starting sentences with "And," "Because," and "But," at the same time, I don't like using "However," "Moreover," or "Additionally." Anyone finicky?

It's a stylistic choice.  And no one is choosing the style you write in.  Do what you want.

Semi-finicky.

My personal philosophy is "who gives a fuck if it sounds cool?" I used to be kind of reserved about until I realized I wasn't in high school english class anymore. I only recently got comfortable with using "And ..." but I try not to do it a lot.

I don't use the last three a lot because I got really self-conscious about empty words after reading The Elements Of Style. Also the book had this to say about the word "however" which I think is pretty sound reasoning.

Avoid starting a sentence with however when the meaning is "nevertheless." The word usually serves better when not in first position.
The roads were almost impassable. However, we at last succeeded in reaching camp.
The roads were almost impassable. At last, however, we succeeded in reaching camp.

When however comes first, it means "in whatever way" or "to whatever extent."
However you advise him, he will probably do as he thinks best.
However discouraging the prospect, they never lost heart.

Ultimately, I agree with Brandon and think you should just do whatever sounds right to you.

It's a good idea to try to use them sparingly, but they are better to use than the formal words unless you're writing in a formal style. Sometimes sentence fragments work well and sometimes they don't.

W. Jordan, you should definitely read through it. It makes a huge difference. The guy is basically like "This is how you write. Period. Do this. Don't do this." etc.

Obviously, you don't have to adopt everything he says, but it is monumentally helpful and makes you aware of a lot of things you may not have noticed.

I remember a book that used the ampersand instead of "and." Forget what it was.

Barr is finicky about the use of finicky.

JGB is proud of his beard - just saying

as for 'finicky' toss it out the window.

Read my lips...

'There is a time and place for everything - literary speaking - saying no only limits your ability to be awesome.

I am proud of my elegantly proportioned cranium and delicate featurees; I'll thank you to remember that. I say to you; good day, sir. Good day.

What's wrong with starting a sentence with 'because'? Seems legit to me. I mean, you still have to wirte a proper sentence (unless it's dialogue), but 

That explains a lot.

I'm always amazed at what kind of stuff I wirte on here when I'm drunk. I didn't even finish (or proofread, apparently) my thought. Because, as we all know, I am an expert on