In A5/1 stream cipher, why are the specific bits in each register chosen for the majority rule and for the XORing step? My two part question is:

1) How are bit positions 8, 10, and 10 chosen for the majority rule for the respective registers?

2) How are bits {13, 16, 17, 18}, {20, 21}, {7,20, 21, 22} chosen for the XORing step for the respective registers, and why the different number of bits for each register?

Were they arbitrarily selected when designing the protocol or is there some advantage to using these bits specifically and this number of bits from each register?
Welcome to crypto.stackexchange. Your second question could be easily answered by a simple search on this site, or elsewhere, but anyway, here goes:

Firstly, A5/1 is well-broken, the non-uniform clocking ends up not helping much, and lots of different initial states merge as the clock ticks--look at the Wikipedia page for A5/1 for a long list of papers with attacks. Now, onto your questions:

Question 1: The majority rule bit positions are not so crucial, as long as they are not from a bit which is being fed back, or from the output bit. Also your observation in the comments: "if you used the same index for all registers, you would get a loop where all the registers shift each time" is incorrect in general. Depending on loading and feedback same any 3-tuple of bits can end up in the majority positions.

Question 2: The LFSRs used are primitive, their tap positions correspond to primitive polynomials, and give the maximum possible output sequence period of $2^L-1,$ if $L$ is the length of the LFSR, a so called maximal length sequence. So the tap positions are crucial. For more on LFSRs and periods, see, for example, the following question which asks what happens when you have non primitive LFSRs. 

With the three LFSRs in A5/1 having relatively prime lengths $19,22,23$, (thus relatively prime periods) the internal state of the cipher has possible size $(2^{19}-1)(2^{22}-1)(2^{23}-1)$ instead of being much smaller. Of course due to irregular clocking not all states need be visited.