Is "da" used often in the casual speech? Or is it often omitted whenever it's possible? For example:
1) Genki (da)?
2) Suki (da)?
3) Ke wa atsui (da) ne?
It is difficult to give a precise answer to this question. In cases where the speaker has a choice between "da" and just ending the sentence, both have their own nuances. Omission may be more "feminine" and addition of da might be more "masculine". In some cases, da can be used for emphasis. Usage patterns vary by gender, age, social situation, and possibly by region/dialect as well.
There are many other casual sentence enders besides just "da"/"" (nothing). You'll also hear stuff like "genki na no?" -> "genki da yo". There's also "kai" and "dai" for questions, "wa", "jya" and many others. In fact, sometimes personal choice of sentence enders is used to "project" a certain "personality type". It is something you develop an "ear for" the more you speak to a variety of people.
Note that there are grammatical limitations, however. For example, as per Yuuichi Tam's comment, you wouldn't use "da" as a question end. For a question, you can replace "da" with "ka" (but this can sound sharp or rude in some cases, so be careful) eg "genki ka?", or just end the sentence with rising intonation "genki?". There's also other question forms like "___ (na) no (ka)?" eg "genki na no?" but that's a more advanced topic.
About the example sentences:
As statements, all three would be okay with da or without. As questions, the first two are wrong if you use "da". The third one sounds weird to me but is close.. if you said something like "atsui da yo ne?" that would probably be okay, but that's because it is more like a statement followed by a request for confirmation: "atsui da yo" (statement) + "ne?" (question)
“だ-da” is a colloquial form of a predicate, "です" - used in both written and spoken form and "である" - used mostly in written form.
“だ” also can be replaced with “だよ,” which sounds softer than “だ.” The feminine version of “だよ” is "だわ" and “だわよ” that you often hear from woman speakers.
You say 今日は暑い（ね）, but should never say "今日は暑いだ" and "暑いだね." It’s odd and ungrammatical.