Will the nebula of Betelgeuse be visible to the naked eye? How bright, how large, how soon, for how long? When Betelgeuse goes core collapse supernova it will leave a supernova remnant. Will it become visible to the naked eye? If so, in what time frame will it be visible. Will it be star like from the beginning, or only after decades or centuries of expansion, to then fade away as it thins out?

The Crab Nebula seems to have had a progenitor of similar mass as Betelgeuse, also a core collapse supernova about 800 years ago, and is about ten times further away. Its supernova remnant seems to be about 300x400 arc seconds in size, so about a fifth of the Moon's diameter on the sky. But it is not visible to the naked eye.

If Betelgeuse's remnant becomes a pulsar, would light from its jets hitting gas be visible to the naked eye, and further out earlier than the supernova remnant itself?
There is an article here that describes the visible effects quite well. In essence, within a week or so, it would be comparable in brightness to the moon and therefore visible during the day. 


  Betelgeuse would then start a phase of final, rapid dimming and again reach its current brightness level after possibly three years. After six years, it would be too faint to see with the naked eye. This would forever alter the visual appearance of Orion and we might need to think of another object the remaining constellation might represent.


This means that after 3 years, Orion would go back to "normal", and after three more years, we wouldn't see it at all with the naked eye - so no, not centuries or decades.

As for the pulsar question, I highly doubt that the jets would be visible... pulsars are very, very dim optically. Their pulses are best observed in the radio spectrum, so even if the emissions did hit gas clouds, the resulting "light" would likely be in the radio spectrum. It would also likely be spinning incredibly fast, since it would be young, meaning that even if there was some optical light from the jets hitting the gas, it would just appear constant to us given just how fast the pulses are - at least with the naked eye, as you asked. 

Now, with other equipment than our eyes, we could likely get some sort of light curve or time vs. intensity profile which would make that pulsation apparent. This would also best be done in radio waves, which are the best bet when dealing with pulsars. (There are a few exceptions, as you mentioned with the Crab pulsar being visible in some other frequencies - see here for more details.)