I pretty sure the answer to all your questions is "no". I'm not completely sure about bertha but I know that the AI doesn't know how to use the silo, so it doesn't build it. Overwhelming nuke defences is much the same.

AI also builds big berthas in front of the base if you allow it to be build. Just like the singularity generators that should blow up 5 Krows each if they attack it =D

Huh... I must be thinking back to SupCom days when I remember being nuked by the AI...

It didn't nuke me at all in my last Zero-K battle. I was intentionally going slow to give it time to build one if it did. I used to turn them off so I didn't have a comp stomp suddenly end because I forgot to build an anti-nuke but I guess I don't have to worry about it... until they surprise me by adding it one day... :)

I guess the other superweapons work because by the time they get built the AI's base has expanded past them.

they use them, but they wont if they see certain things like an anti-nuke

AI can use Berthas, Zenith, Tactical and regular Nukes as well as DRP but it may choose not to.
I had a 2 hour long 1v1 battle vs Brutal AI at a Gecko Island map, Brutal attacked me with nukes when I had an antinuke, murdered my defences with a DRP but I blew it up with Likhos and it also tried to use the Zenith but fired at a remote extractor base on the edge of the map, after which I blew it up as well.

So yeah, AI can use them but may not see a reason for it.

Interesting, I'll keep an eye out then. I've not built an anti-nuke in my last 4 battles against brutals and cutting edge AIs and haven't seen any nuke buildings.

I regularly see the AI build Diso Rave Parties when I don't block them.

Short answer, yes.

They will build these things IF you let them and if the difficulty is high enough. The first two difficulties not seem to build them at all, but everything higher than that will build them. The hardest levels will build Berthas regularly and generally they use them to finish the game.

Same with Nukes, however, they do scout and if they see everything covered by Anti-Nuke they will go with an Artillery option.

I wish they used the planet carver. Now that would be cool.

They use the Starlight as well!
Had it happen on Twin Lakes redux. Brutal AI used it after ~90 minutes.