Yeah, tager is pronounced as tar in normal spoken Danish and the computer voice is not too far off here. Infact, you'll sometimes see it written as ta'r when the text refers to something someone said. The apostrophe is used like in English to mark that some letters have been left out.
It's a kind of slang and quite colloquial Danish — unlike in English where it's a normal practice. We don't have many words that get this treatment, others that come to mind is hva' (hvad), ska' (skal), ha' (have), ik' (ikke).
So you might read Hej, hva' ska' du ha'? in a novel, meaning Hi, what do you want/need? (a shop assistant in a bakery could greet you like this).
The "d" at the end is pronounced and the computer voice does a good job at it here.
But Danish is a funny language and I can understand if it doesn't sound pronounced to you. However, having the "d" makes a difference -- I would read the fictional word "afste" with stronger stress on the "e", and so the "d" serves to moderate the vowel before.
In other words, there is an aspect of look-ahead in the Danish language. Better examples would be "lige" (straight or just) and "ligge" (lie down) where the "gg" in the latter word makes the "i" vowel short. You can hear the difference on http://www.ivona.com/.
Taking leave can be used and suggest an action taking place in the present tense. You could getting ready to exit a party and say "I'm taking my leave" or be asked "are you taking your leave" and it makes sense. It wouldn't be the same thing as saying someone is "On Leave".
Not sure if you noticed but your answer is quite different from Bruno's. You said 'taking insert-possessive leave'. The possessive is compulsory when you are talking about leaving a place. It makes a big difference.
Also I wasn't contesting the use of tense in Bruno's answer, so not sure why you brought that up.
Tense or otherwise the English translation, which is the example you gave, still is applicable as I’m registering what you’re saying. So I guess what I’m saying is your example flatly could confuse a non-English speaker who is also learning Danish. It bares saying that the meaning of that statement from either principle position can mean either or regardless of “is”, both would be applicable. No harm meant, no harm caused.
I simply wrote the woman leaves and it is accepted. But I've read your previous post above and can't agree more with you. You are right, in my language the litteral translation is "the woman takes leave" and it means exactly that, better than the woman is leaving or the woman leaves. But the problem is we must use an intermediate language (english) and sometimes the english sentence means something only in english.
See the example given above : "She takes off". It may be perfect english, but for me it can only mean a plane is leaving the ground or a woman is completely drunk, nothing else. If I had to translate it in danish the problem would not be to say the woman leaves but to figure what the english sentence means.
It is not a complaint, it is a very good thing because it is the opportunity to greatly improve english, and incidentally danish.
Imagine "at tage" also means "to go" except it HAS to be used with an adverb that describes where you are going or you can use the preposition "til" which equates to "to" and is used the same way. This means you can actually say "at tage" and then just add an adverb at the end to describe where you are going "at tage afsted/derhen/derop/hjem/til spanien" (to leave/to go over there/to go up there/to go home/to go to spain). You can only use certain adverbs that describe some kind of position though
If you really want something that means "to leave" directly the closest would be "at gå". It literally means "to walk" but we use it when we're talking about leaving as well. E.g. "Jeg tror at jeg går nu" meaning "I think I'll leave now"
For Dutch speakers, this could be a good 'ezelsbruggetje', (=mnemonic) : 'Tager afsted' kind of looks like the Dutch (literally) : 'Nemen afstand' So 'afstand nemen' Translated into English, this means: 'to take distance'. I hope this helps in any way!
(Disclaimer) I don't know if 'afsted' and 'afstand' are valid cognates. This just helps me remember the word and I'm just sharing my ezelsbruggetje.