Can we omit subject or some verbs in Chinese?
Yes you can, let me first give the possible short version of the example you gave:

What is your name?
FULL VERSION: 你叫什么名字？/你的名字是什么？(What's your name?)
SHORT VERSION 1: 你叫什么？ (Here we don't usually say 叫什么？ as its tone is more rude or intimidating. To make the tone softer, normally you could add 呀ya/啊ah at the end such as 叫什么呀？And it also could mean &quot;why the yelling&quot; if used in wrong context, so I don't recommend using it)
SHORT VERSION 2: 你叫……? (Your name is...?) actually it's more &quot;you are called...?&quot;
SHORT VERSION 3: 什么名字？(What's the name? Bond) OR 名字？(Name?)
WRONG: 什么? (什么 means &quot;what&quot;, it would be too general, the listener would have trouble to understand what you are asking)
WRONG: 叫 This is not a valid short version of &quot;what's your name&quot; because it doesn't give any hit to the listener about what you are asking, and it could be misunderstood as a command to shout or yell.

要去哪里？ yào qù nǎlǐ? where are you going?
SHORT VERSION: 去哪(儿)？

吃饱了. chī bǎole I'm full. [This one is correct]
饱了 bǎole I'm full. [This one is correct] moreover it could be both a question or the answer depending on the tone. I.e. Q: 饱了？(Are you full?) A: 饱了。（I'm full)

When spoken, the grammar of Chinese is very flexible. Except in formal settings (e.g. broadcasting, meetings, court, press conference), you can almost bend the grammar in anyway you want as long as it makes sense.
By making sense, I mean that the listener should be able to understand and fill the omitted part in the meaning with minor effort.
[Below is some additional stuff, if above is enough for you, you can skip the rest]
Actually, as a Chinese, when I was in middle School (sort of 7-9th grade), after learned the grammar, we spent more time focusing on constructing sentences without ambiguity rather than practising the grammar. I mean to eliminate ambiguity, you must know some grammar of the language, but for Chinese, it's more important to avoid ambiguity rather than to construct grammatically correct sentence (sometimes, grammatically correct sentences just doesn't make sense and not so grammatically correct ones just makes so much of it). I guess this is determined by the fact that Chinese is an analytic language.
Moreover, sometimes, ambiguity is preferred in circumstances such as when the speaker want to convey two meanings in one sentence with a tweak of certain words. But if you tried to be clever and failed, people will think you are pretentious.
Anyway, in everyday life, when you want to omit something in Chinese, just make sure it's understandable and there is no ambiguity, you'll be fine.