If I'm not wrong you can have one project open at a time in the main kicad interface,
I can have any number of projects open that open in separate instances of KiCAD. A project is stored in its own directory in the file system and there is a project file (.kicad_pro), schematic (.kicad_sch) and PCB (.kicad_pcb) etc. within each directory. The schematic and PCB files are locked when opened, to warn you if you try to open the same file multiple times. I haven't run into problems copying and pasting parts between different projects. As far as I have observed, it works no differently than copying and pasting within the same project.
Yep, as I said. For schematics, no issue at all. Just be careful in v7 if you have enabled automatic annotation. If not, the pasted symbols will be pasted without annotation. If it is enabled, I would be wary. I had tried and it seems to retain the original annotation rather than create new annotations on the fly, but to be checked, haven't tried thoroughly - automatic annotation has given me more trouble than benefits so I disabled it.
Also, I've later realized that the question seemed to be more about copy-pasting layout blocks specifically, and in that case:
- Any graphical item not connected to any net: absolutely no problem
- Footprints/traces/anything connected to a net: it retains the original annotation and net names. So if annotation and net names are the same in the layout in which you are pasting, that will work. Otherwise, that will wreck havoc. So, you can use it to copy-paste already layouted blocks as long as annotation and net names in both the destination and source schematics were exactly the same for the copied block. Also, you can't do this if you have already imported the corresponding footprints in the PCB editor from the current schematic, as it would create duplicates when pasting. So, requires a lot of care. There are plugins for this, but haven't tried them yet. I have used the "Replicate layout" plugin, which is meant to replicate the layout of identical hierarchical schematics inside a single project. It mostly works, a few quirks that sometimes require manual fixing afterwards, but pretty handy if you have multiple instances of the same schematic block.