You are correct, of course, but can you change voltage without changing current? I draw a specific current through the base to get a certain voltage.
No, that base current is waste leakage current that comes from the charge carriers that do not make it from the emitter to collector, and are instead lost to the base due to the voltage on the base. It is a phenomenon that can be minimized, but not eliminated. Ib is exponentially proportional to Vbe, as is Ic also exponentially proportional to Vbe. Therefore, Ic and Ib are linearly proportional to each other within a range of currents. This proportion is called beta. That makes Ib a indicator of Ic, but it is still Vbe that controls both Ic and Ib. The Ib is a nuisance current that has to be taken into account.
With the same base voltage and current can I have different collector conditions?
When you say base voltage, you do not mean Vbe, do you? What I am saying is that if you set a current on a BJT with no Rb or Re and note the Vbe, the Vbe will be the same if you add Rb and Re and adjust the base voltage to show the same Ic.
I cannot calculate collector voltage knowing base voltage, or can I? I can take collector current, divide it by gain and get base current. Can I calculate base voltage know collector voltage?
Since a BJT is a reasonable current generator when operated in the active region, the collector voltage will not influence the Ic very much. That can be seen from the Ic vs Vc curves. Since betas vary considerable between BJTs unless they are selected, it is not a good idea to rely on beta for design. Anything can be calculated if the parameters are measured and known, but that is not always practical.