Yes, vacation days count towards duty days and need to be substracted from the year total. With 22 vacation days you end up with 178 days and that also includes all recurrent and ground training! Your net flying days would therefore be something like 160 to 165 days a year. If you volunteer of extended days, you work more, but you obviously also make more.
I never took part-time, so I cannot comment on it with full confidence. As a new joiner you will probably not be able to apply for FW (flexible work patterns) for the first year, or two. But I might be wrong on this! There are several FW-programs to have you work more or less than the 200 standard days a year. People who are on a reduced FW-program will work normal tour-lengths and then have longer off-periods between them. Keep in mind that summer months MAY be excluded from this and you will just work less between October and May. But I am not an expert on this part of the contract/policy.