Wheezy,
Correct on both one and 2. However the fact that the starter drives the lower inertia H.P. spool is an additional benefit, but the main benefit is that each compressor spool is better matched to the airflow so in theory there are increased stall/surge margins over a twin spool engine. Surge margins at different power settings are however maintained by the use of engine bleeds and variable intake guide vanes on both twin and triple spool engines.
A couple of disadvantages of triple spool engines (in theory at least) are increase in weight, increased complexity and therefore higher lifetime maintenance costs.
Regards
Exeng