Is it enough for crossing/ascending/descending IFR-flights to adhere to the separation minima?
This might not be the answer to your question, but maybe it will help you understand separation. 

Your question specifically asks for how separation is ensured between climbing, descending and crossing traffic over navigational fixes (NAVAIDs).

Flight routes are planned for traffic to use certain flight levels en-route, which follow the semicircular/hemispherical rule in effect in that country or are bound to the airway being used. In my example picture below I will use the German semicircular rule of eastbound flights using odd flightlevels. 

(Image Source: Own Work - Program used: Aircraft Situation Editor (ASE))

Both aircraft will cross the HMM VOR roughly at the same time, based on the ground speed displayed. The westbound aircraft (DLH123) is at FL200 (even FL), the eastbound aircraft (BER456) is at FL210 (odd FL). The separation between both aircraft overhead HMM VOR will reduce to 0nm lateral separation, but 1.000ft of vertical separation, and thus, the separation criteria are met.

(Image Source: Own Work - Program used: Aircraft Situation Editor (ASE))

In this example, I need BER456 to descend to FL100, but I have crossing traffic from the right which would cause a cleared conflict, if I issue a descent clearance directly to FL100 without any conditional instructions. I can use a bit of mathematics here to ensure separation however and issue a clearance nevertheless.

Both aircraft are travelling at 420kt GS. An aircraft will also descend 1.000ft in 3nm, rule of thumb from every flight instruction book available. To ensure 5nm of lateral separation, I would need a buffer of 2.000ft during descent to the crossing traffic, not factoring in that the other traffic will be on a divergent track anyway. Let's just measure the distance to the HMM for simplicity.

My instruction to BER456 would be:


  R: BER456, when ready descend FL100, cross HMM VOR at FL230 or above (rate of descend 2.000ft/min or less)


With this instruction, I can issue a clearance to BER456 that will allow the aircraft a continuous descent to FL100, while ensuring minimum separation is kept to the other traffic. Overhead HMM VOR, both aircraft will have 3.000ft of separation. After BER456 has descended a further 2.000ft, it will have 6nm and 1.000ft of separation, making it well clear of a conflict, especially since the traffic is divergent.

The dirty solution would have been to clear BER456 to FL210 and after 5nm issue another clearance to FL100, but this doubles the frequency time and also interrupts the descent for BER456.