This might sound arbitrary, but if someone said いすがテーブルのとなりにあります it would sound like just an observation, so more like "a chair is next to the table." But with いすはテーブルのとなりにあります it puts emphasis on いす so it feels like "THE chair is next to the table." Know what I mean? But they are both correct translations.

agree, confused by when to use the different structures. I also replied using that structure as it was used in one of the previous phrases [there is a chair next to a table = テーブルのよこにいすがあります] but after that it only accepts "A" は "B" の location あります

I am struggling with the same thing I never know what order to put things in when の is involved

Most cases, "yoko" and "tonari" are interchangable. The difference is that "yoko" can be used with different types of things (like plants next to dogs), and "tonari" can't (must be dogs next to dogs). If I remember correctly. The problem may be the use in kanjis.

For indication of existence:

あります is for things, plants

います is for human, living creatures such as animals, fishes.

となり means "neighbour" so you would use it when 2 things are right next to each other, without anything in between, or at least not anything of the same sort.
よこ means "horizontal" (it is a noun though). You would use it when 2 things are in the same row.

隣(となり) is used when two or more things are of the same kind, quality or status (you cannot use it when you are talking about objects next to people or viceversa); also, it is used to refer to the closest thing next to an object, regardless of the distance. Also, 隣 is used to refer to neighbours or neighbouring houses.

横(よこ), on the other hand it is used simply to mean "next to", just like となり, but not caring about kind, quality or status.

Taken from here: http://yesjapan.com/YJ6/question/4683/soba-_tonari-_yoko_what_039-s_the_differen

Because となり is used when items of the same category and size are next to each other, with nothing between them. In this case, the chair and the table are not the same size.

as you probably know, の is a particle that shows posession. So テーブルのとなり can be thought of as "the table's side" I guess.

Support the answer above.

The same usage as in テーブルの上、テーブルの前 which should look more familiar to us.

First of all, 'the chair' is the subject. So the sentence shouod technically start with いすは. Think of wa as putting more emphasis on the preceding noun whereas ga is less specific. (The chair vs a chair)

Secondly, when using a particle like に(ni) to denote location, it usually comes right before the verb . (にあります)

As you have it written it wouod read something like, "Next to table a chair is" It should be, "The chair, next to the table it is'

looks correct to me.

i don't think so... grammatically, the が and に particles are typically used with あります and います. the が would also make the point of emphasis (subject) of the sentence the chair, which seems right.

It is tonari.. Not donati. Difference between tonari and yokoni had been explained as above. Hope it helps..

One little mistake can change the entire meaning of a sentence. Take a comma, for instance. It the difference between "Let's eat, grandma" and Let's eat grandma. Sometime a "little" mistake can have big repercussions on a sentence.