The simple answer is yes. You can start a sentence with "and" and be correct. However, it can make your writing more effective if you try to avoid "and' and use one of the other adding conjunctions listed in my article. Why? "And" is easy and sounds a lot like our typical speech, but when you spend the time to think about which other sentence starter fits, you often get a more nuanced meaning in your sentence. "And" tends to connect two ideas equally but does not always show the relationship between those ideas. Consider the following example which uses two coordinating conjunctions "and" and "but" to start sentences:
Anna went skiing yesterday on the steep run at Whistler that I warned her not to try. And she made it down the hill just fine. But then she slipped on some ice at the bottom of the run and twisted her ankle so badly she can't ski today.
Now look at a re-write which uses sentence starters:
Anna went skiing today on the steep run at Whistler that I warned her not to try. Moreover, she made it down the hill just fine; however, when she got to the bottom, she slipped on some ice and twisted her ankle so badly she can't ski today.
"And," "but," "or," and "so" are all conjunctions which join two sentences together, or explain the relationship between items in a list. Many people are taught not to use them at the start of a sentence, but the truth is that many of us use them all the time when we are speaking, and the Chicago Manual of Style suggests that up to 10% of written sentences start with one of the coordinating conjunctions. Coordinating conjunctions also include "yet," "or," and "nor" and are often remembered by the term FANBOYS (for, and, nor, but, or, yet, so). Final answer? It is not improper or wrong to use coordinating conjunctions to start a sentence, but it may not be the most effective technique and is often overused by less experienced writers. Additionally, since many people have been taught, it is incorrect, others may judge your use of "and" to start a sentence with poor writing and incorrect grammar. Therefore, I tend to tell my students not to use a coordinating conjunction to start sentences if they can avoid it.