Yes, HIV does attempt to prevent multiple infection of the same cell. Once a T cell or other cells are infected, HIV downregulates the CD4 receptor which it uses to gain entry. Because of this, there are less receptors so another HIV is less likely to infect that cell. Meanwhile, the infection itself upregulates CD4 receptors on other cells which means the downregulation is even more effective. There are other mechanisms but this is a major one.
Hiding means it is somewhere where we don't know where it is and it isn't that active. When HIV is active it is destructive but also the cell is aware it is there. The cell can then try to alert other cells that it is infected. If HIV isn't active but instead hiding, it can stay where it is without really being detected. Even the drugs we use usually only affect HIV whilst it is replicating. So again if it's in a cell that's long lasting and the HIV is not replicating then it is essentially hiding. Such a cell includes macrophages and memory cells when we are just talking about the immune system.
Will HIV reside in the lymph node? Well yes, this is an exciting place for it. There's loads of exposed CD4 receptors and lot's of it's favourite T cells about. An infected cell will transmit HIV to other cells when they are signalling to each other. Even worse is when a dendritic cell is infected. Instead of instructing the immune system how to detect HIV infected cells, it is infecting them.
HIV is an infection. As soon as the body detects it, it goes into overdrive trying to eradicate it. This means there are more cells for HIV to infect. More foreign material means the immune system is even further stimulated to make even more cells. However it doesn't want CD8 cells to proliferate too much nor too many HIV specific cells so driving too much proliferation is a bad idea. What HIV does do is make lots and lots of useless copies of itself. HIV particles that are so badly made they aren't functional. The body tries to fight these and makes more cells to combat these but there's no point because they aren't the problem. But this immune diversion and a larger target number of cells is exactly what HIV wants. Proliferation of HIV specific cells, particular CD8, is associated with delayed disease progression or control.
Usually the cancers caused by HIV are because of other viruses running unchecked. Kaposi's sarcoma or lymphomas for example.