I think the firmware updates are automated, and they probably stagger the process during non-peak times so less people are affected and less at the same time to reduce possible call volume to customer service. It's possible that you need CMTS access to manually update, which only a few people have for security reasons.
As for Tivo, Cox doesn't support the actual Tivo device, just the cable cards (CC) and tuner adapters (TA) needed. As for how long Cox will support cable cards, that is complicated. First there is the difference between how long they will stock and give new customers CC and replace current customer's CC for troubleshooting and how long current working CC will keep working. I think many Cox stores have stopped having CC but you may still be able to have one mailed to you. YMMV depending on where you live. However, many ISPs (like Comcast) have stopped people from new installs of CC so I think it's just a matter of time before Cox makes that rule national. As for them not working, I know of 3 separate issues. One is the power key issue. This is a chip on some CC that is bugged (think Y2K bug) that is used for time keeping. I think those stopped working around October of last year. This isn't just a problem with people with Tivo and CCs because most of the boxes themselves also have CC and work in a similar way. Next, many ISP are changing from QAM to IPTV. This is a similar change to when many phone providers switched from circuit switch tech(AKA POTS) to packet switch tech(AKA VOIP). This is done so there is more bandwidth available for internet service. Since IPTV doesn't work with CCs, it won't work with Tivo. Last, Cox is upgrading some areas from cable (HFC) to fiber (FTTP). This means you use an optical network terminal (ONT) to connect to Cox's network instead of a modem/gateway. Since the ONT doesn't have a coaxial output, there is no way to get the signal/data to the CC. This is all supported by the fact that the FCC doesn't mandate ISP to support CC anymore and it's more profitable not to support them because it pushes more customers to boxes, which make them more money.
TL;DR While I haven't heard of a deadline by Cox, the writing is on the wall. Probably a good idea to start transition to a different service. If you don't want to pay for boxes but still want to use Cox TV service, you might want to look into Apple TVs that are compatible with the Contour app. If all else fails, there are streaming TV providers like YTTV, Fubo, and DirectTV.