Indeed - six connections: PWR, GND, RESET and the three SPI wires. Could create a block of six plated holes in the PCB and then you press pogopins in it to make the connection. A regular Arduino can be the programmer; I've used a Pro Micro for that purpose. The problem is that connecting LEDs to the same pins will likely mess up the signals.
You have 12 I/O of which one is the RESET, so effectively 11.
Seven for the display, reserve three for the SPI part, one button (a push button can use an SPI line safely).
Then you also need two for the RTC (I2C - those are normally shared with the SPI lines - should have no problem with programming but not enough experience to say for sure).
I/O:
7x display.
1x push button (to wake up the processor and have it start displaying the time, one digit at a time).
2x I2C.
I2C and the 11th pin are also the SPI (programming) interface. An ATtiny24/44/84 or the 441/841 would be suitable. The latter appears to have I2C slave built in (have to read the datasheet better - it'd have to be able to act as master and slave in this situation).
To be sure you can keep the time, an RTC can come with battery backup. That adds a small battery to your device, but at least you don't have to reset the clock (unsure how - maybe some trickery with Serial or via the I2C bus which would be available through the programming contacts) every time the supercap runs dry. Assuming it will now and then.
This is quite a project. Selecting components (and studying data sheets), building a prototype (going straight from design to PCB is of course possible but risky - very hard to make corrections), write software, design & manufacture PCB, getting all components. The final assembly is the easy part, an hour or so of soldering. The first part can easily be 20-30 hours of work. Not building the first prototype saves about 1/3 of that.
Come to think of it, I wouldn't want to send out a bag of parts and software and hope it goes fine. I'd want to have the PCB assembled, software installed - otherwise if something goes wrong it's almost impossible to debug and get working.
Wouter.