The time machine in Balthazar's basement is probably not actually the Epoch but the "Neo-Epoch" he mentions. Note that it only has one seat instead of three.

There are no Nu in Chrono Cross, but there are weak monsters called Beach Bums that sort of resemble them. One of them can be found in the Neo-Epoch's room providing the service of changing the lead party member's name, just like the Nu in the Laruba Ruins in Chrono Trigger. Cross has plenty of other references like these, including an enertron machine and Leene's Bell, plus the ones Ash has already mentioned.

And yes, Cross is definitely a direct sequel to Trigger, at least in the sense that its events are directly caused by its predecessors. Chrono and company's triumph over Lavos turns out to be the main cause of Serge and company's problems, and so on. The FMV endings added to the PS version of Trigger had scenes that were intentionally designed to help tie the two games closer together; besides the one showing the theft of Masamune, there's also one in which Lucca finds Kid as an orphaned child.