My questions are:
1) Does this filter have a name? ie Colpitts etc.
2) How can I calculate the required inductor and capacitor values for the above bands?
As G0HZU has noted the designer of this has missed a low value series decoupling capacitor before the base of the amplifier circuit. It will likely be similar in size to c4 so that it doesn't load down the filter.
As far questions go:
1. This is a cascaded Bandpass filter (ie in this case 2 parallel tuned circuits set to the 20m band ~ 14,2MHz). If you could afford another inductor its better to avoid the low value of C4 and place a series tuned circuit in this leg in order to make a bandpass PI filter. This will improve insertion loss. Regardless what is presented works and is a commonly used version of this filter.
2. Use the parallel tuned circuit equation
ie 1/(2pi*sqrt(LC)
the equivalent series capacitance of C8&C13 = C12 = 27pF
C8 & C13 exist because the designer is trying to convert from the low antenna impedance (maybe 50Ohms) to the high impedance across the top of the tank circuit (maybe 10K -dependent on Q of your inductors). ie the reactance of C13 is much lower than C8 so provides better power transfer into the tuned circuit from the antenna. Most receivers of this variety use an input transformer instead (a Toko KANK was once common but these are now hard to get) but this arrangement also works.
IF you are going to use a small telescopic whip antenna to listen to local stations you would be better to remove/replace C8 and C13 with a single capacitor of the same value as C12 (27pF) and then attach the telescopic whip to the top of this capacitor (ie not the ground side). This would provide a better match than the low impedance point set up by the capacitive divider.
For a similar reason C4 is determined experimentally in order to not lot cross-load the tuned circuits and will be very small 1-10pF. As it is a very low value it will make this tuned circuit somewhat lossy so its generally experimentally adjusted as high as possible in order to maintain selectivity (reduced loading) while maintaining a decent insertion loss.
Probably the most widely built version of this class of receiver (they all look much the same) is G3RJV's sudden
https://archive.org/details/73-magazine-1991-10