Once you reach large insulae housing/oil requirement, do you still need to supply pottery and furniture to those houses to maintain and climb from that level or just oil (but then furniture again if you fail, etc.)?

Also, what role does entertainment play in this chart?
https://gaming.stackexchange.com/questions/282520/housing-steps-and-requirements-in-caesar-3

As for your first question, no, you have to continue supplying everything they needed up to that point. So it goes
food --> food + pottery --> food + pottery + furniture --> etc...

That chart lists the required entertainment value needed to reach and maintain the corresponding level of housing. I don't think you can actually read off those entertainment values ingame. It's calculated behind the scenes and is a combination of a house's direct access to entertainment (i.e. when last a gladiator has walked past for example) and the city's total entertainment coverage. This FAQ explains it in question 12:
http://caesar3.heavengames.com/faqs/faqs3.shtml

Thanks. I was having some problems maintaining pottery and furniture levels supplied from a distance in the fifth peaceful mission, and one block was also requesting more entertainment instead of pottery which was a bit confusing. A couple of warehouses set to getting pottery didn't seem to do anything while others did.

[img]https://i.imgur.com/BpmR3op.png[/img]
https://i.imgur.com/BpmR3op.png
Pottery is grabbed from the east, and the block northwest of the screen here was supplied fairly well, while the southwestern block wasn't at all despite the way being as clear.
The market seen here was a last effort to get some pottery southwards (this save is from shortly before I finished the mission), there are two more per block as well.

No problem :) I believe that the warehouse set to 'get goods' sends out a single cart guy to a warehouse with pottery, and comes back with 400 (?) units. Thing is, as soon as there is pottery to take somewhere at a warehouse (that is not set to 'get goods') the cart guy sets out to fetch it, and if it's not there when he arrives then he returns with nothing (actually I have to test this I'm not 100% sure of it). So a serious problem can then arise if the source warehouse of pottery runs out periodically because a cart guy might arrive at the moment it is empty. But as long as the pottery source remains stocked at least most of the time then all the warehouses that are set to 'get goods' should get enough (unless one warehouse serves too a large population, but at one a block that won't be the case). So if one of your warehouses struggles to get enough pottery then I think your pottery industry is just not big enough. Or, the warehouse which is meant to act as the source of the pottery is frequently raided by market buyers because it's closer to the market than another you have set to 'get goods' from it.

By the way I notice you have fish in the warehouses. Just in case you didn't know, the market buyer only collects food from the granary. What will happen is that the warehouse worker, aka cart guy, will periodically take the fish (100 units at a time) to a nearby granary.

Just out of curiosity I looked at my old save for that level. I saw that I was importing both clay and pottery, and I managed to produce enough pottery with 6 workshops:

I see. I actually waited very long to start importing pottery as I kinda screwed up early on (almost lost to various disasters and an invasion due to building too much industry too soon) and put it on hold to save money, so it might've gotten better in a few more years or with one more dock.

The fish was just me forgetting to not accept meat at that warehouse, I still keep forgetting stuff like that.

Very nice! I'll try going for villas in the next mission.