If an infection is severe the adaptive immune system, T- and B-Cells, has to be activated during an immune response. Otherwise, the disease would quickly overwhelm your innate immune system.
To answer a): No. When thinking of cells you have to think big. There are 25 million to a billion different T-cells in your body and up to 10 billion different B-cells and all of them have different antigen receptors. You could say we are protected from every antigen/infection, the right T- and B-Cells just need to be activated for that.
To answer b): No again. When an infection becomes too severe the Dendritic cells gather information on the enemy by "covering themselves" in the antigen. Then they journey off into the lymphatic system to find the T-Cell that recognizes the antigen. This takes a while but remember, there are many, many Dendritic cells trying to find the T-Cell that recognizes the antigen, not just one. The T-Cell then multiplies and the T-Cells "split" into two groups, one goes to the site of infection to help fight off the enemies while the other group goes out to find the B-Cell that recognizes the antigen. This journey is a bit easier than the journey of the Dendritic cells since the B-Cell has already been "half-activated" by the antigen flowing in the lymph. The B-Cell has already multiplied a bit and has already started making "ok-ish" antibodies. The T-Cells can use that to their advantage and follow the antibodies. The T-Cells then meet the B-Cells and fully activate them starting a complicated process that refines the antibodies produced by the B-Cells.
To answer your final question, without B-Cells there would be no antibodies and the fight against the infection and the fight would be much harder. Without the T-Cells there would also be no antibodies since the B-Cells would never be fully activated. The B-Cells would only get "half activated" by the antigen flowing in the lymph. Half-activated B-Cells die after circa 24 hours to save resources because T-Cells only activate if the infection is too severe. Also, remember the part where the T-Cells split into two groups? The group that goes to the site of infection helps the innate immune system fight the enemy, so if there were no T-Cells helping against a dangerous enemy, the innate immune system would be overrun.
Note: This is a very simplified version. For more reading, I recommend "Janeway’s Immunobiology", or "Immune" by Philipp Dettmer.