It is not so much about mortar as it is about the working process. You should clean out the old crumbling mortar to make place for the new one. Remove the old mortar so that all parts are firm and nothing is crumbling - you can even scratch out a bit of the old mortar. The very important part is to wash the wall properly with water so that all parts hold firm and there is no dust - otherwise the new mortar would just fall out. Than press the new mortar in the joints and fill them, make them smooth on surface.
As for the mortar the lime content should not be important for the lifetime, but you can add a bit for better workability. Generally there should be no or very little lime in a stone wall. The new mortar should consist of 1 part of portland cement and 2-3 parts of sand.
If you do it properly (well prepared surface, good mortar), the new mortar should not crumble or dust for tens of years.
A personal experience: it is a very time consuming monotonous work, but the result can be very good if done with care.
It depends...
These days, bricks are used far less frequently as the "structure" of residential homes, due to the higher chance of failure from lateral movement (such as an earthquake). Depending on the construction of your home, they may provide little to no structural value and may be purely cosmetic. If they are cosmetic, the structure should not be compromised any more than it was when the mortar failed (which would typically mean "insignificant"). If they are structural, your concerns may be warranted.
If your house was built more recently, uses primarily wood framing including along the interior sides of your exterior walls, and your rafters/trusses are fastened to said framing (rather than fastened to the bricks), it is more likely to be cosmetic. It is extremely telling in the case where the brick stops before it reaches your roofing timbers. That is not to say the brick provides absolutely no structural value -- it still serves a purpose and can help resist movement, but cosmetic brick does not bear the structural loads. If you've seen a home built with brick recently, you likely saw a free-standing structure, covered with plywood and a waterproofing/resistant/wicking/other material, before the brick "walls" were added.
One of the biggest differences between cement and mortar is typically the lack of aggregate (sand) in cement. If your mixture contained sand, you probably used the mortar as intended. The sand exists, essentially, to give the cement something to stick to, just like rocks do for concrete. Without the sand, I suspect that the cement will eventually loosen, crack, and fall away much sooner than mortar; it also will not prevent movement of the bricks in the same way mortar does, because the "connection" that the provides is missing.
If you cannot determine that your brick is solely cosmetic, I would recommend that you remove the cement and replace the bricks using mortar. When in doubt, consult a professional about your specific situation.
Best Answer
As long as it is dry, you are good to use it. If it has been exposed to excess humidity in any temperature it will harden in the bag.