My Chinese teacher says she doesn't care. But for future reference, is 个 acceptable as a measure word for all nouns?
No, of course. Here's a summary about the kinds of cases where 个 is inappropriate.
sth slender. e.g. 一根绳子, 一条蛇, 一条路; 一线光明, 一绺头发, 一丝细雨.
sth thin. e.g. 一张纸, 一面红旗, 一幅画.
sth solid in certain kinds of shapes. e.g. 一块砖, 一团面, 一坨泥.
sth amorphous. e.g. 一杯水, 一壶油, 一斤米, 一碗沙子.
sth forming part of a greater whole. e.g. 一层楼, 一重山, 一级台阶, 一节甘蔗, 一段木头, 一截电线.
sth where the measure words come from verbs or nouns and can't be replaced by 个 without risk of changing their meaning. e.g. 一捆柴, 一把米, 一撮毛, 一抱草, 一包糖, 一任县长, 一束花, 一串糖葫芦, 一堵墙, 一服药, 一堆土, 一封信, 一滴水, 一朵花, 一本书, 一篇文章, 一瓣花瓣.
And note that even if 个 could be used instead, the meaning might be different. e.g. 一位老先生 (respectful) vs 一个老先生 (not necessarily rude, but not respectful).
At last, this might be an opinion issue; I agree that they're understandable in daily conversion; the listener will try to supply the appropriate measure word. But IMO the abuse of 个 as measure word is not acceptable for Chinese learner, especially who want to master natural Chinese.
Reply to @lly Your example(three head of cattle) is not appropriate, because English and Chinese are belong to different language families. For example, if I say tense is useless, since I can say 我昨天回答了. Why do I need to change the form of verbs? Obviously my predicate does not make sense. I should not use a language to judge another language.
You given another topic to discuss, what OP asks is Can we use 个 as global classifier, yours is that the classifier is redundant(only for being knowledgable)
1. Is the classifier redundant?
Linguists have found that a language won't have plural forms and classifiers at the same time. That is a interesting fact.
English does not have quantitative relation forms for syntactic category, so it does not have classifiers only measures.
Classifiers are related to gender of noun, it classify the noun.
2. Is the 个 acceptable for all nouns?
Check below, 个 is common word, but in some special scenario, in my story I don't know the form of the object, 个 does not give me enough information, I don't know how many I need ( I think it is small). It is the least functional classifier.
================= original answer
No, but somebody does, which makes you confused.
A true story:
Once I went to a drug store for kalium permanganicum(disinfectant), after I said what I need, the saleswomen asked me "你要几个?", cause I had never bought it before, I didn't know what does it looks like, I replied: 可能需要几个. (Maybe several)
Then she found some little box, I said: 一盒就够了... 这论个吗?(one box is enough, is this classified by 个?).
That is the first moment I realized the purpose of classifier, I still remember the scene even several years passed.
譬如以每一个整体作为单元，“个”确实是很常用的量词。但更多的情况下，量词决定了描述者的认识意向，也影响了接受者的感觉。以水为例。用容器作为量词描述的很多见，一口水（a mouthful of water），一碗水，一杯水，一瓶水，一罐水，一桶水，一车水（水车承载）等等；也有直接使用物理单位来描述的，一升水，一吨水，一公斤水；抑或以形态或其他形式描述的具体的水，比如一滴水。
I think this is one of the things that trouble many people who are learning Chinese.
There is no single rule about the usage of 个. Sometimes, you do need to use it. Sometimes, you are actually forbidden to use it. @songyuanyao already gives you an excellent answer. I am going to give you some examples to illustrate the point abuse of 个 as measure word is not acceptable for Chinese learner, especially who want to master natural Chinese.
The examples are about time units, namely, days, months and years.
In English, you say three days, three months and three years. No need to have measure word.
In Chinese, you say 三天 (three days), 三个月 (three months) and 三年 (three years).
You don’t say 三个天, 三月 and 三个年.
In fact, Chinese would think you mean March when you say 三月.
It’s just the convention. I really don’t know any rules about this.