Translation:Put the armchair here, right under grandmother's portrait.
Yes, it works. The noun with «над» is always Instrumental but за and под are used with Accusative when you mean direction.
When talking about putting objects somewhere, these two can be interpreted two ways: moving object to intended position or make the action of putting "there" (though, I think в/на strongly prefer Accusative in such situation). In this sentence «под портрет» sounds a bit funny to my ear, so I would probably not use it it my speech. However, when an object really has a place "under" it, it would be OK. For instance, putting a box under a table or putting a folder under the portrait when said portrait is lying flat on a bed or a desk.
By the way, I don't know if you're interested in feedback on your English, but "I would not probably use it" sounds a bit odd to me. I would probably not say that. Or at least not write it, it's the kind of thing I might say when talking and not taking so much time to think about it.
Yes, «под портрет» would be also correct. At least it sounds equally good to my ear.
The word "the" (before "grandmother's") is definitely not required here, and as someone pointed out, it is only used if you are referring to the grandmother of someone you don't know.
Missing the word was accepted, but I was told that what I wrote had a little error (being the "the"). I'm reporting this now.
On a different note, is there a strict rule of thumb for when to use the definite and indefinite articles in Russian? In English, the sentence "Put an armchair here, right under the grandmother's portrait." would be correct, given the right context. The current tree does not accept this as a solution...
To be honest 'the Grandmother's portrait' sounds terrible to me. The only situation I can think of where it sounds fine is if it is someone else's Grandmother and you don't know that person. If you and the people moving the armchair are not related to the Grandmother in question but know she exists then you might well tell them to put it underneath the Grandmother's portrait, but if you're speaking about your own Grandmother or the Grandmother of somebody else present then you'd be far more likely to use a possesssive pronoun than an article (e.g. put it here under my Grandmother's portrait/under your Grandmother's portrait).
If I were speaking about putting any (arm)chair, I'd rather say "Поставь сюда кресло" ("Под портретом поставь кресло").Well, I'd generally state the place first. Otherwise you HAVE to change your intonation or you will sound as if there was a certain chair you want there.
Probably, you can think of it as follows: if the object is specific, you tell where it is or where you want it (alternatively, you may say that it is THIS object that is there). If it is just one of a group, you "fill" a place with an instance of this type, which often calls for a "there is" structure in English and moves the object closer to the end of a sentence in Russian.
Stylistically neutral word order will generally start with things known about the situation and then move on to things being said. Thematic place and time will generally be near the beginning (unlike English!) If the subject and the verb are both part of your message, verb goes first, which you can clearly see in "У меня есть" sentences (the subject is the thing you have, and «есть» is the verb).
"Put the armchair here, right under the grandmother's portrait" is unnatural. It would have to be someone else's grandmother, not connected with either of the parties, and that's pretty weird. It's not strictly wrong but "Put the armchair here, right under grandmother's portrait" is a better answer
No, it also works. With sitting, standing, lying or placing objects, both interpretations are fine(in principle).
Well, I would definitely prefer "Поставь молоко в холодильник" to "...в холодильнике" but the latter is also grammatically correct (it just draws unnecessary attention to "some place inside the fridge").
В предложении "поставь кресло сюда, прямо под портретом бабушки" не согласованы первая и вторая части предложения. Первая половина предложения отвечает на вопрос "куда", а вторая, которая должна уточнять первую, отвечает на вопрос "где". Более естественными, согласованными, выглядели бы варианты: "поставь кресло (где?) ЗДЕСЬ, прямо (где?) под портретОМ бабушки" или "поставь кресло (куда?) СЮДА, прямо (куда?) под портрет бабушки".
My translation with help of Google Translator:
In the sentence "поставь кресло сюда, прямо под портретом бабушки" the first and second parts of the sentence are not consistent each other. The first half of the sentence answers the question "where to?", and the second, which should clarify the first, answers the question "where is?". More natural, consistent, would look like the options: "поставь кресло (где?) ЗДЕСЬ, прямо (где?) под портретОМ бабушки" or "поставь кресло (куда?) СЮДА, прямо (куда?) под портрет бабушки".
Sent report about unnatural Russian sentence.