Question about usage of measure words
I understand that measure words are used whenever you use a numeral, but I'm a bit confused when it's used in a sentence where it doesn't seem necessary to me (though I'm not Chinese of course, so there might be a logic to it that I am missing completely :D). For example, when I say "I'm a teacher", I've seen it translated as 我是一名老师, instead of just 我是老师. Are the numeral (一) and the measure word (名) absolutely necessary in this case, or is it just a style thing? Does it sound strange if I leave out the numeral &　measure word?

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我是一名老师 -- wrong translation
我是老师 -- correct one

measure words are used where the numeral is used or the quantity is essential\

meanwhile there are plenty of words which need no measure words, eg 天

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我是一名老师 is more offical  for example, if u give a speech to present urself or some other condition u should say 我是一名老师   however   if someone ask about ur profession, u can simply say 我是老师 他是学生 我父亲是司机 etc.  this is just for conversation witch is not offical.

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In the example you gave the word "a" serves as an indefinite article rather than a numeral so you can emit it when you translate it into chinese but when the word "a" serves as numeral you must use chinese numeral in the chinese translation too
For example:I have a cat. you must translate this way 我有一只猫 rather than 我有猫

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有点难回答。
怎么说呢，这个要看不同的场合，看别人怎么问。你的情景太短，我不能简单的回答是不是可以省略“一名”。
我是老师 是最简单的表达，你加入形容词，量词等等，可以使句子表达的更清楚，意思更明确。

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“我是一名老师”的完整表达更为正式一些；“我是老师”是比较随意的表达，口语中甚至可以说“我是一老师”，与英语“I am a teacher”刚好对应