Can you wire a T12 fluorescent Strip light and a T8 fluorescent strip light together?
Those aren't strip lights; that's something else.  

If you like your old fixtures, or don't want to deal with the ugliness of a swap, keep them.  Every part of a fluorescent fixture is serviceable.  If the ballasts are fried, you can just do a ballast swap.  You can also convert to LED.  All within the old fixture. 

The fixture is the expensive part, so swapping ballast or going LED will be the cheapest repair, and lets you keep the quality fixture instead of dealing with the absolute junk fixtures I see in big-box these days at the $20-30 price point. 

If you want to switch to T8, the T12 and T8 tubes are physically compatible, so T8 tubes will fit in a T12 fixture. They are not electrically compatible, so you must change the ballast to T8. 

You can also get LED "tubes" that snap into T12 or T8 sockets.  Get ballast-bypass (direct wire) types which are fed from opposite ends of the tube.  Then rewire your fixture to bypass the ballast.

The 4' and 2' fixtures will be compatible, they share only 120v mains power and have separate ballasts.