Electronic – Switching between battery and USB using diode OR logic

Your circuit concept is starting in the right direction but it requires some important changes for it to work.

You will require series base resistors for all of the PNP transistors. As drawn you will cook them with no base current limitation.

Secondly the NPN transistor is drawn backwards. Its emitter needs to connect to the GND.

Third you need to do some serious work reorganizing the way the bases of the various transistor are biased. You need to separate the 12V and 5V PNP bases so that the 12V one can shut off when the base drive goes away to the 5V one. After this fix it is necessary to provide also a separate pullup of the NPN transistor.

Don't cringe. Resistors are incredibly cheap.

Lastly this schematic, to be bluntly honest with you, is horribly drawn being that it does a bad job of following conventional standards for how a good schematic is drawn.

I'm going to hand wave over a lot of detail here, but the short answer is that your "b" is correct. You effectively have a race condition and in order to answer exactly what happens you need to know about the propagation delay of your logic (and if we want to really dive into the nitty gritty wire delay etc). You would need to know how the 3 bit counter is implemented (e.g. with which gates and how those gates are implemented) and the propagation delays of the gates. You can think of the propagation delay of the gate as the "delay" of the gate. For example, if you take an inverter and toggle the input there's some delay till the output stabilizes. Take a look at this data sheet and search for "tphl" and "tplh". Notice that the two propagation times are not equal.

It is not necessarily true (in fact it is very probably necessarily not true) that all outputs of the counter toggle at the exact same time. Some outputs (depending on the implementation the more significant bit outputs) will require more logic and thus probably have a greater combinatatorial path to path to output (read more propagation delay). Furthermore there is no reason to assume that the inputs to the counter start at 3'b000. With these two facts the output of the counter will run around like crazy (the shortest combinatorial path way will gate the max output frequency e.g. the shortest combinatorial path will limit how fast the fastest output bit changes). Your display won't be able to keep up with the counter and will effectively display "random" numbers (obviously what the display actually shows depends a lot on its implementation).

For a simplistic, but similar example google "ring oscillator" (but don't build one unless it's for educational purposes only!) and "propagation delay of logic gates".