The individual rules that have been mentioned by your teachers are correct; however, you have to pay attention to the priorities. For any given compound, there may be other relevant rules (e.g. concerning the principal characteristic group that is expressed as suffix, or the longest chain) that determine the parent chain and the numbering of locants first. Only if there is still a choice, the rules concerning multiple bonds are taken into account.
In this case, the principal chain is the chain with the greater number of multiple bonds. If there are more than one chain with the greatest number of multiple bonds, the principal chain is the one with the greater number of double bonds.
The corresponding rules in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) read as follows:
P-44.4.1 If the criteria of P-44.1 through P-44.3, where applicable, do not effect a choice of a senior parent structure, the following criteria are applied successively until there are no alternatives remaining. (…)
The senior ring, ring system, or principal chain:
(a) has the greater number of multiple bonds (P-22.214.171.124);
(b) has the greater number of double bonds (P-126.96.36.199);
Likewise, with regard to numbering of locants, low locants are given first to multiple bonds, all considered together as a set. If there is still a choice, low locants are given first to double bonds:
When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
(i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
(ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
The “lowest sum of locants rule” that you have mentioned does not exist in the IUPAC Recommendations and can lead to wrong results. The corresponding section in the current IUPAC Recommendations actually reads as follows:
P-14.3.5 Lowest set of locants
The lowest set of locants is defined as the set that, when compared term by term with other locant sets, each cited in order of increasing value, has the lowest term at the first point of difference; for example, the locant set ‘2,3,5,8’ is lower than ‘3,4,6,8’ and ‘2,4,5,7’.