What capacitor type is best for low tolerance and thermal stability?
The old Wax Paper capacitors  For those who have never seen one, you can get an idea of their size by the size of the resistors. The large brown cylinders are the capacitors  : 



The tolerance on the capacitors wasn't so good.  And they were huge physically. Not the best tolerance on temperature either.   

You can safely choose among any capacitor that has same voltage rating, capacity, tolerance, and temperature dependence, when it comes to replacing old Wax Paper capacitors.  

(Unless you are a super audiophile) winky winky.  

EDIT 1  :  Robert Endl pointed out in a comment that perhaps the capacitor might be Mica. In the photograph mica capacitors can also be seen.  They would be the blocks that appear with color dots on them (similar to resistor color code). If indeed the OP's capacitor turned out to be a mica capacitor,  they are often now days called "silver mica" capacitors.  If the capacitor is a mica, then there are silver mica capacitors available.  

EDIT 2  : With new photo from OP  :  The two red capacitors on the left do appear to be Mica capacitors. Just to understand Mica capacitors,  they are very high Q ( high quality) capacitors that have very low loss. Very temperature stable too.