Could someone give any good explanation of why there are some compositions with "de" seemingly at random?
For example we have:
"de pe masă" - "on the table"
"de la tine" - "from you"
And of course, "de lângă mine" here. Seems like "de" often acts in combination with other prepositions, but there's no information of that. Is it completely wrong to omit it? What kind of combinations require it?
"de pe masă" is a "complement circumstanțial de loc". If I translate "complement direct" as direct object then the above thing would be circumstantially object of location (??). It determines whatever pronoun, noun is on that table/location. The same with "de lângă mine". I wish I would have paid more attention to my grammar in school so I could help you now :) . Maybe someone will give a better explanation.
It's hard to answer these questions because from what I remember from my school days, we were supposed to learn the rules (or "the how") by heart. There wasn't really any explanation provided for "the why."
Such explanation might not even exist. Why does one say in English "I had eggs for breakfast." and not "at breakfast?" Did I have the eggs for the breakfast's sake? You know, like when I say "I am doing this for you." :) Prepositions are complicated.
Momzi mentions "circumstantially object of location". Well, I found some notes about "locative prepositions", such as, "din", "de jos", :de sub", with sample constructions ("hainele DIN dulap" - "monstreii DE SUB pat") and my dictionary cites "deasupra" somewhere. I "know" nothing more at this point, being a measly A-1 amateur. Formidabil!
Everything is random until we find its logic. Everything is confusing and complicated until we understand. Romanian has a really tough grammar, but don't worry, you'll learn.
When 2 or more prepositions are together they become another preposition ( it is called prepoziție compusă) with a different meaning. In your example, "de pe masă", you need to treat "de pe" as a preposition which is neither "de" nor "pe", but in the same time is "de + pe" because "pe masa" still means "on the table" and "de" here means "from", but because in English no one says "from on the table" and "from the top of the table" is too long, we translate "de pe masă" as "from the table". Have you understood anything or it is more confusing than before?
Because "women" is plural, while "is" is singular. You need to have both the subject and the verb agree in number. So, for the plural form it would be:
"The women next to me are my wives." = "Femeile de lângă mine sunt soțiile mele."
Also, I see that you used "next to me" while the English translation for this sentence uses "beside me" and it might not be set up to accept "next to me" as an equivalent translation.
Hmm, I'm actually not sure that there's a difference between "beside me" and "next to me," but perhaps a native English speaker can speak to that. I would translate "de lângă mine" as either one. In Romanian there's also "alături de mine" which could be translated as "by my side."