How does bonding between non-complementary bases occur?
How do the two bases that aren't complementary remain bonded?


Non complementary bases can only form non-Watson-Crick bonds, which are very unstable and noticeably different.
Therefore any mismatch in the DNA of the genome will form a recognisable disruption of the DNA helix. This disruption is usually removed very quickly by the cells, since it makes DNA much more unstable.

Most cells have multiple mechanisms to detect and repair these sites, with one system specifically dedicated to DNA mismatch repair.
This system acts on mismatches directly after DNA replication, since at this point specific markers (sometimes DNA methylation, sometimes not fully understood) can differentiate between the newly synthesised DNA strand (which may contain an error) and the old one.   

The repair mechanism will remove the mismatch and replace it with a properly bonded base pair, so that mismatches in the DNA are always short term issues - it is still possible though that 'wrong' mismatching base is replaced, which then (finally) leads to a mutation.

The error rate of DNA polymerase (1 in 10^7 sounds about right, this is the combination of the initial error rate and the DNA polymerase proof-reading mechansim) therefore has to multiplied with the error rate of this repair mechansim, which is around 10^3 (I can't find a source for this right now), to get a final error rate of DNA replication in cells.