"Поставь кресло сюда, прямо под портретом бабушки."

Yes, «под портрет» would be also correct. At least it sounds equally good to my ear.

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.

It can mean either.

Still not accepted: 23/06/2018

Why is "the" before grandma's portrait not required, atfer all, articles are necessary for English nouns. (I understand that they are necessary not before all nouns, but "grandma's portrait" is another case).

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).