Will radar/lidar still work when every car is equipped with them?

You'd be surprised.

This is actually topic of ongoing research, and of several PhD dissertations.

The question which radar waveforms and algorithms can be used to mitigate interference is a long-fought over one; in essence, however, this breaks down to the same problem that any ad-hoc communication system has.

Different systems solve that differently; you can do coded radars, where you basically do the same as in CDMA systems and divide your spectrum by giving each car a collision-free code sequence. The trick is coordinating these codes, but an observation phase and collision detection might be sufficient here.

More likely to succeed is collision detection and avoidance in time: simply observe the spectrum for radar bursts of your neighbors, and (assuming some regularity), extrapolate when they won't be transmitting. Use that time.

Notice that wifi solves this problem inherently, much like described above, in a temporal fashion. In fact, you can double-use your Wifi packets as radar signals and do a radar estimation on their reflection. And since automotive radar (802.11p) is a thing, and the data you'd send is known to you and also unique, you could benefit from the orthogonal correlation properties of a coded radar and the higher spectral density and thus increased estimate quality of time-exclusive transmission.

There's a dissertation which IMHO aged well on that, and it's Martin Braun: OFDM Radar Algorithms in Mobile Communication Networks, 2014.

This is a rather old problem in radar engineering, dating from the era of jet aircraft carrying guns and supersonic missiles. This Wikipedia article on Chirp compression gives some clues about how the problem might be equally addressed at automobile speeds.

There are military sonar and radar systems that see the world around them using reflections of "the other guy's" radar / sonar. They existed back in the days of 286 Intel processors... so it can be done much more cheaply today when a $5 ARM SOC is as powerful as a 1983 Cray XMP-48 (the machine I managed then...)

So while it is useful to use all the time domain and code domain multiplexing, it is also possible to compute the location of the other emitter and then use his signal to see the world around you.

I know this existed in the 1980's as I knew the engineer who built it for the military and visited his shop. It was secret then, now not so much.

Basically, call multiple emitters a "feature" and move on.