1. I'm not sure if I remember the procedure correctly, but there were special magistrates (the censors) who gathered information on citizens. How much land and/or money they owned etc. According to that, they were registrated and according to that, they got told in which part of the army they had to serve (Accensi, Hastati, Principes, etc.) since in the Pre-Marian army, the soldiers had to pay for their own equipment. So it would have been impossible for a small farmer to serve as Princeps since that gear was quite expensive (shield, mail, weapons). The Patricians were nobility whose wealth was based on owning land. I'm not sure but I think they're were not allowed to participate in commerce. The Equestrians were the bourgeoisie so to speak. They were not nobility but quite wealthy since their wealth comes from commerce. Especially in the Late Roman Republic they were becoming increasingly filthily rich, against some of those people today's super-rich would be mere beggars. The Equestrians had to serve as cavalry since they could afford horses. This is quite different from feudal systems where usually the nobility provides the cavalry. But Roman cavalry never really was something prestigious.
The citizen army of the Republic was organised like a militia. Citizens would participate in regular exercise and in the event of a war, they would be levied.

2. The Patricians were the landowning nobility. The were not responsible for protecting their land specifically. I'm not sure about the taxing. For some offices it was necessary to be a patrician in order to be able to assume it (like the consulate, if I'm not mistaken).

3. Yes, non-patricians were allowed to own land. In fact the bulk of the (citizen) population were free farmers. Being a farmer was a very prestigious profession. The crisis of the Late Roman Republic has to do with farmers being unable to take care of their land because constant wars kept them away from their land. A great many of them impoverished and had to sell their land cheap to wealthy patricians who accumulated enormous estates making them even richer. The farmers who sold their land would then lease it from the patricians. Thus the foundation of the army deteriorated and there was a lot of social tension. There have been many attempts to introduce new laws limiting the amount of land one person could own but they were all foiled by the short-sighted patrician opposition (some quite violently... the attempts of the Gracchi for example).