A couple of dozen tribes across the whole world - that isn't many. I would suggest going with more than that. If you are only looking at a couple of dozen tribes, they are extremely unlikely to ever meet. I imagine you might want them to be able to to trade and have conflict. Even if you wanted the tribe you started with to believe they were the only ones left in the world, you might still need higher numbers for them ever to run into anyone else.
Looking at the numbers
I don't have any experience with mathematical modelling, but here is a very rough approach to help you think about it:
Current world population is 7 billion. Suppose there is 100 people to a tribe, 70 tribes, and so 7000 people in total. The population has reduced so that it is now only 1 per million pre plague.
Plague kills 99.9999%
1 person per million survived the plague, leaving 7000 people in the world. It might be extremely difficult for the survivors to actually find each other, band together and start repopulating the earth.^ Many plague survivors would not be capable of surviving on their own until they joined a group (eg children). Some survivors would not be capable of reproducing.
Manilla is the worlds densest population at 1.5 mill/35 km - there would have only been 2 survivors in 35 km. Delhi would have 11 survivors over 400 km. Most people would not have a survivor closer to them than thousands of kms away.
So I'm not actually sure if humanity would be able to come back from this just because of the sheer geographic spread of survivors! But they might be able to. They would definitely be able to if they could group together.
On the plus side, with this survival rate you wouldn't have to think up any extreme scenarios causing more mass casualties.
^ I guess that people could communicate by radio. And set fires to attract notice.
Plague kills 99.999% of people
One in 100,000 people survive the plague, leaving 70,000 people spread across the world. Individuals would still find it difficult to band together and would be more vulnerable to fatalities until they were part of a group. (Manilla would have 70 survivors over 35 km bad, which seems quite reasonable.)
Plague kills 99.99% of people
One in 10,000 people survive the plague, leaving 700,000 people spread across the world. Again, individual survivors would have to survive on their own for a period (and more prone to fatality by themselves), but it wouldn't too extreme. You'd still need to kill off most of the population though.
Plague kills 99.9%
One in a thousand people survive the plague, leaving 7 million people spread across the world. There would have to be a fairly big secondary cause of deaths to reduce the population down to 7000 in a few generations.
Plague kills 99%
70 million people survived the plague. That is way too many. Barring some other really significant apocalyptic event (like a volcano exploding and making extreme changes to the environment), I can't imagine how the population could reduce to 7000 in a few generations.
Secondary Deaths/Population Bottleneck
Regarding secondary deaths - I think what you're talking about here is the risk associated with a population bottleneck.
"In consequence of such population size reductions and the loss of genetic variation, the robustness of the population is reduced; the ability of the population to survive selecting environmental changes, like climate change or a shift in available resources, is reduced."
Regarding some of the examples of secondary deaths you provided, here's my 2c:
Starvation: Surely canned food would be abundant for all the survivors for the first few years/generation at least. Might be a problem for small groups later on when non farmers have to figure out how to farm, unlikely to cause mass casualties (eg reducing population from 7 mill to 7 thousand).
Accomodation: would be abundant for all individuals/groups.
Weather: Would be an issue for individuals at the beginning. Extreme weather events would also be an issue for some groups, but weather events would be unlikely to cause mass casualties unless there was a volcano or something that threatened the entire world. If there was a secondary climate change angle that did cause mass casualties, then it may explain why society chooses to be tribal/nomadic - to show their a respect for nature.
Rotting corpses: My gut feeling based on watching lots of zombie movies is that corpses can be fairly easily be moved out of the living zones to avoid contamination. Getting clean water would be the hardest thing, especially for lone individuals, as it would rely on their intelligence/knowledge/resourcefulness. This would only be a problem for the first couple of years, and I imagine most groups would be smart enough to figure out a way around this.
Rabid dogs: All of the bands of rabid dogs would probably be a big problem for survivors to have to deal with - especially lone survivors.
Pandemic: Further viral outbreak could definitely cause more casualties. But would you want to go there in a big way twice?
Lack of health care: Would increase the mortality rate, and slow the baby boom. A bigger concern in small groups.
Nuclear spills: I previously said that this would be geographically isolated, and wouldn't cause mass casualties. I think now that I was wrong. There is a good chance that the Northern Hemisphere may be in a lot of trouble. When the power (and then back up generators) stop, nuclear powerplants storing spent fuel rods outside of the core would explode. (There are many hundreds of these in the Northern Hemisphere especially North America, Europe and Japan). Smoke containing highly radioactive fission products would circle the Northern Hemisphere for a few weeks, and then precipitate out in the rain. I'm not sure what exactly this would mean for survivors, but it could be a pretty big cause of mass casualties. It would not be as bad in the Southern Hemisphere because there are a lot less nuclear powerplants.
Deaths due to fighting: I imagine violence could be fairly common, but I think the type of violence that would cause mass casualties would be pretty unlikely.
Interesting snippet: Humanity may have bottlenecked at ~2000 for more than a hundred thousand years pre stone age. So there is precedent for the population not to remain static at those numbers for some time.
Minimum viable population
"An MVP of 500 to 1,000 has often been given as an average for terrestrial vertebrates when inbreeding or genetic variability is ignored. When inbreeding effects are included, estimates of MVP for many species are in the thousands. Based on a meta-analysis of reported values in the literature for many species, Traill et al. reported a median MVP of 4,169 individuals."
I imagine the individuals included in the 4,000 number would have to be a certain geographic distance from each other to prevent inbreeding traits. (ie, they would need to be close enough to breed).
"A slightly different form of a bottleneck can occur if a small group becomes reproductively (e.g. geographically) separated from the main population, such as through a founder event where for example a few members of a species successfully colonize a new isolated island."
Real life examples:
Tristan da Cunha (current population 264) was populated from eight men and seven women.
Pitcairn (current population 50 after extensive emmigration) did the same with 15 men and 12 women.
Both islands are extremely isolated and survived for 200 years. Only Tristan da Cunha had problems with inbreeding and they were relatively mild.