Sounds interesting, although I feel like point 2 is a bit too restrictive.
Right now, it seems to require public availability of the program’s source code, even if the program is only used internally. This, in my opinion, is too restrictive and would make organizations not want to touch software under this license, even if they are friendly to the FOSS community and want to make the end result open source. After all, during internal development, they might be experimenting with lots of different programs/libraries to find the one that best suits their needs, modifying them if necessary, then tossing out whichever one isn’t required. Having to then also put out a public repository during that step feels like too much (and could also hurt secrecy – competitors could see "oh, this company is now publishing a source code for libfoo, they must be working on some foo-related project).
Perhaps a better rule would be something like “if you deploy the program, you must make the source code available in such a way that anyone who has the technical capability and legal right to use the program can also receive its source code”. This would, of course, mean that while the program is being developed and tested internally, the source code likewise only needs to be shared internally, but when it becomes available publicly, the source code has to likewise be available to any potential user (meaning that someone doesn’t need to, say, buy or own a device running some firmware to get its source code, like with GPL).
Typically yes, a free / open-source software license gives the user the right to use the program, to spread it around, and to create derivative programs based on the original source code, as long as the new derivative program follows certain rules.
For permissive licenses (such as the MIT license), the rule is something like “put this acknowledgment somewhere the user can see”, if you’ve ever seen an “Open-Source Licenses” menu item somewhere, that’s where they usually put these.
For “copyleft” licenses (such as the GPL), the rule also requires that your modified version is also distributed as open-source under the same conditions.
Most definitions of free or open-source licenses also require the license to impose no additional restrictions on how, or for which purposes, the software can be modified. The RAR archive extractor, for instance, is not open-source, because the license for its source code prohibits it from being used to make a program capable of making RAR archives. Some developers add conditions to restrict undesired uses, such as commercial or military use of their software. These licenses are likewise not considered “open source”, but rather “source-available”.