Learn Hardcore Chinese: zhè shì shénme?

Use 是 to equate/identify with a noun. 这什么？ is not standard Mandarin (you might hear dialectal/clipped forms like 这啥？, but avoid them as a learner).
Note: With adjectives you typically don’t use 是: “It’s big.” = 它很大。 (not 它是大。)

What’s the difference between 这 and 这个 in this question?

Both 这是什么？ and 这个是什么？ are fine.

这 = “this” (pronoun).
这个 = “this one” (uses the default measure word 个), often slightly more colloquial or contrastive. When modifying a noun, you need a measure word: 这本书, 这个人, 这张票.

How do I say “What is that?” and “What are these/those?”?

That: 那是什么？
These: 这些是什么？
Those: 那些是什么？

How do I pronounce each word and what are the tones?

这 (zhè): 4th tone (falling).
是 (shì): 4th tone.
什么 (shénme): 2nd tone + neutral. The me is very light. Tips:
zh and sh are retroflex; curl the tongue slightly back.
The vowel in shì isn’t “shee”; it’s the syllabic “sh” sound with a short falling tone.

Why isn’t there a 吗 (ma) at the end?

吗 is for yes/no questions. With a wh-word like 什么, you don’t add 吗.

Yes/no: 这是书吗？
Wh-question: 这是什么？
Yes/no with 是不是: 这是不是书？

How do I answer this naturally? Do I need measure words?

Typical pattern: 这是 + [number + measure word + noun].

这是一本书。 (本 for books)
这是一个杯子。 (个 as a default)
这是一张照片。 (张 for flat objects)
这是一台电脑。 (台 for machines) If it’s uncountable or a proper name, no measure word: 这是水。/ 这是北京。
Without a classifier, 这是书。 states the category; with a classifier (一本) it feels like “a single book.”

What does 这是什么书？ mean compared with 这是什么？?

这是什么？ asks for identity.
这是什么书？ asks for the type/category: “What kind of book is this?” Similarly: 这是什么语言？ (What language is this?), 他开什么车？ (What car does he drive?)

How is 这是做什么的？/ 这是干什么的？ different from 这是什么？?

这是做什么的？/ 这是干什么的？ asks about purpose or function: “What is this used for?”
Structure: [做/干 + 什么 + 的] modifies an omitted noun (like 东西).
Example: 这是做饭的（工具）。 “It’s used for cooking (a tool).”

Is it okay to say 这个是… instead of 这是…?

In everyday speech, 这个是… is common, especially when contrasting: 这个是新的，那是旧的。
In careful writing, many prefer 这是… unless you need contrast/emphasis.

Are there casual or regional alternatives for 什么?

Yes. In northern/colloquial speech, 啥 (shá) is common:

这是啥？ ≈ 这是什么？ Very clipped: 这啥？ (not standard; avoid in formal contexts).
You may also hear softeners like 这是什么呀？

Why do some people say zhèi for 这?

Before a measure word, 这 (zhè) often surfaces as zhèi in casual speech:

这个 → often pronounced zhèi ge.
In 这是什么？ there’s no measure word after 这, so keep zhè.

What’s the traditional-character version?

這是什麼？ Pinyin/tones are the same: zhè shì shénme?

Can I add particles like 呢 or 呀 to change the tone of the question?

Yes:

这是什么呀？ sounds curious/friendly.
这是什么呢？ can sound reflective or inviting explanation. Meaning stays the same; particles adjust tone/attitude.

What about 这是什么东西？ — is that different?

东西 means “thing.” 这是什么东西？ literally “What thing is this?”
It can add emphasis or mild surprise/negativity depending on tone. Neutral in some contexts, but it may sound a bit blunt.