How does the grammar of 拿把伞 work? This sentence is taken from lesson 2 of HSK3 and apparently it means "get an umbrella". 

The full sentence is 我下楼去给你拿把伞 which should mean "I will go downstairs to get you an umbrella".

My IME is always suggesting 那把伞, which obviously means "that umbrella", which makes me think this construction might not be as common ... ? 

So my question is if this "拿把伞" really is 拿(verb) 把(measureword) 伞(noun). Because I have never seen a measure word attached to a verb in this way, at least not in Chinese. Can all transitive verbs do this? Are there other verbs that behave in this way?
帶把槍 bring a gun.
切條魚 cut a fish.
吃碗麵 eat a bowl of noodles.
裝盒餅乾 pack a box of cookies.  

Here measure words (better see them as classifiers) are 把, 條, 碗, 盒 where the number of one is usually omitted. If it's more than one, the number 2, 3, 4... would not be left unsaid or unwritten. (And interestingly 把, 條, 碗 imply singular and 盒 implies a group which might be plural.)

As for your questions in the comment however, when the classifiers are put at the beginning of a sentence, the number is always present even if it's singular. People don't leave classifiers at the start of a sentence without a numeral.

一盒餅乾在桌上 a box of cookies is on the table.
兩碗飯盛好了 two bowls of rice has been filled / prepared.
三條魚跳出水面 three fishes jumped out of the water.
四把槍在門口 four guns were at the gate.

Again even all of the above examples are singular. The number of "one" would always be described because the classifier is at the beginning.