As far as I recall, the secondary in an H-bomb is indeed compressed by the fusion boosted fission of the primary but this then initiates a fission reaction at the core of the secondary where there is a fission "spark plug". So the lithium deuteride fuel is caught between the incoming compression driven by the primary and outgoing explosion of the secondary spark-plug.

When the fusion fuel burns the resulting neutrons are used to fission the tamper of the secondary and it's this fission is the main energy source for most H-bomb designs.

Yes, inside the fusion fuel capsule there is typically a small amount of fissile material. The fusion capsule is compressed through the very strong forces from the ablation of the capsule's outer material. This compresses the fusion fuel as well as the "sparkplug" inside. The sparkplug doesn't need to contain much fissile material compared to an ordinary fission bomb core because the high degree of compression makes it easier to achieve critical conditions. The sparkplug explodes as a fission bomb, injecting heat energy into the fusion fuel capsule and driving up the conditions (temperature and pressure) which cause fusion reactions. It also injects neutrons which kickstarts the breeding of Tritium from Li-6 and Li-7.

Sometimes the "pusher" around the fusion fuel capsule is also made of fissile material which gets compressed, or imploded, just like a normal fission bomb core, but with much higher forces and so much greater efficiency and effectiveness.

Additionally, while Uranium-238 is not normally fissile it can be fissioned with high energy neutrons. The "problem" with U-238 is that the energy range of neutrons that induce fissions only partly overlaps with the natural energy range of neutrons produced by fission reactions. Meaning that on average a given neutron emitted by a U-238 fission will cause fewer than 1 additional fissions, so it can't self-sustain a neutron induced fission chain reaction (which, I guess, is a good thing, otherwise nuclear bombs would be nearly trivial to make). However, fusion reactions generate very high energy neutrons, essentially all of which can cause fission in U-238. For this reason many thermonuclear weapons have a natural or depleted Uranium casing. Fission in the casing typically doubles the yield of the weapon compared to the yield from the primary/secondary fission/fusion reactions.

All of this can be dialed to suit different desires for bomb production or yield. For example, the "Tsar Bomba" was a 100 megaton 3 stage thermonuclear warhead design, it was tested without a fissionable Uranium outer casing to avoid production of nearly 50 megatons worth of fission product fallout over the test site (which was in Russia), as a result it exploded at around 50 MT yield, almost all of which (97%) was from fusion reactions. On the other hand, you have a device like the W-88 which is an incredibly compact warhead carried on US SLBM submarines. It is a thermonuclear warhead but you could also look at it as a two stage fission-fission device with a heavily fusion boosted secondary, as most of the yield comes from fission in the secondary and the casing.

To give a feel for the forces involved, I have read [2] that the pressure generated by ablation that compresses the W88 fusion secondary is around 6 petaPascals.

For comparison, the estimated pressure at the Earth's center [1] is around 360 gigaPascals, about 1/16,000 that of the W88 secondary compression.

[1] https://en.wikipedia.org/wiki/Inner_core

[2] https://en.wikipedia.org/wiki/Thermonuclear_weapon#Comparing...

edited to add W88 implosion pressure link