1. Linguists believe that Proto-Indo-European had no definite or indefinite articles. Some of its daughter languages, like Latin and Sanskrit, did the same. However, many modern Indo-European languages have both. A few branches still don't, like the Slavic and Baltic languages, and Hindi/Urdu & Punjabi. Some other Celtic languages, like Irish Gaelic, also only have definite articles. (I don't know if there are any Celtic ones that do have indefinite articles).
2. Polynesian languages have both.
3. Many have neither: Chinese languages, Japanese, Korean etc.
4. Semitic languages, like Arabic & Hebrew, only have the definite article.
5. Some languages have more than just definite and indefinite articles, like French and Italian, and more than one set of definite or indefinite articles expressing other meanings as well.
6. It is common for highly inflected languages, often without articles, like Latin and Proto-Indo-European, to evolve into less inflected languages over time. As they rely more on word order than inflection, articles tend to evolve.
Definite articles typically evolve from demonstrative determiners (this, that, these, those, etc). Indefinite articles from words that mean "one". Such is the case with Romance languages like Spanish, French, and Italian. Sometimes the same words are used to mean one/a/an, without distinction (as in French, Italian, etc).
*** Yes, there are many languages without an indefinite article. There are many more with neither an indefinite article nor a definite one. There are also some with additional types of articles.