To understand this problem, you need to understand the linguistic concepts of 'bound' and 'free'. To understand the concepts of bound and free, you have to understand the idea of a morpheme. A morpheme is an abstract linguistic unit that everyone knows how to use, but no one really knows how to define: 'dog' is an English morpheme; so is un-, or -ly. Dog can be used in a wide variety of places in a sentence; un- and -ly cannot. These two types of morphemes are thus called 'free' and 'bound'. Chinese morphology is similar to English in this basic sense. Of course there are important differences too. There are a number of books that discuss this, I recommend Jerry Norman's 1988 book, Chinese.
Here are two examples that illustrate the properties of bound and free in Chinese. First is the character 椅. This is a typical Chinese bound morpheme. Any Chinese-English dictionary will tell you it means 'chair'; what they usually don't tell you is that you can't just say 買一張椅. 椅 is bound; to use it as an independent word you have to at least use the compound form 椅子. Note it does not require the suffix 子. It can be used in other compounds perfectly well without this, both on the right side: 躺椅, 藤椅, 電椅, and on the left side 椅墊, 椅背, 椅套. This means 椅 is not like English affixes, such as un- and -ly.
Compare this with 床 'bed'. Unlike 椅, 床 is used by itself: 買一張床 is a perfectly good sentence. It can also be used in compounds: 吊床, 單人床, 床墊, 床頭, etc. 床 is free; it can be used as an independent word.
How do you know which characters (morphemes) are free and which are bound? This is what I think the OP really wants to know. I don't completely agree that the difference between free and bound is a continuous spectrum. In different dialects, a morpheme may have a different status: a bound morpheme in dialect A may be a free morpheme in dialect B. A morpheme's status may also change; many morphemes that were free in early texts are now bound in modern Chinese. But the Mandarin dialects are largely consistent in what is free and what is bound, so free and bound are well worth learning for a foreign student.
Absurdly, there is NO modern dictionary that marks characters (morphemes) as bound and free. There WAS one which you might be able to find if you are near a large library: The Dictionary of Spoken Chinese, published by Yale University's Institute of Far Eastern Languages in 1966. This was based on an earlier work by Chao Yuen-ren and several colleagues, compiled during WWII. Although very old, for the common words that bother every student, it is a great resource for serious students.
I agree that sometimes the free-bound distinction can be tricky. Going back to the question, 爸 can be used by itself, but mostly as a vocative form (when you are talking to Dad); otherwise it can be odd: 他是我的爸 is odd for many people, 他是我的爸爸 is normal for everyone. The status of 爸 as a free morpheme is limited; you are best treating it as a bound form, except in talking to Dad.
白 is free, but only with the meaning of 'white'. In the adverbial sense of 'without paying' or 'in vain', it is like all Chinese adverbs: bound.
把 bǎ has several meanings. As a structural particle it is bound, but a special kind of bound.
As a morpheme meaning 'handle' it is emphatically bound. Be careful, there are actually two morphemes meaning 'handle': bǎ, as in 把手, and bà as in 刀把. This is probably a historical change, there are some forms where there are two pronunciations: eg 把子 can be bǎzi, or it can be bàzi. I've heard both. What I haven't heard is 買一個把. So for standard Mandarin, 把 is bound.