I'd say my process was a few months from the initial application and offer. The Culture Fit assessment is just looking to see if you will be a good fit within the Amtrak culture (of safety). This test is relatively new as it is part of a new initial online assessment. As long as you have a mind for safety and a bit of common sense you should pass that just fine. After that there generally is a written testing session. Details and some sample questions along with a Realistic Job Preview will be sent to you. For mine, it was in a small group session at the crew base I applied to. They explain a bit about the position, pay, benefits, and expectations (basically extra board hours, nights away from home, the assigned day off isn't always guaranteed, etc...) Once you pass the test a "fit test" will be ordered at a local contracted facility. This basically tests your arm and leg strength. There is an example of the test on the Amtrak Careers website. Assuming a good result on that, next step generally will be an interview where you are asked a series of standardized questions. If you make it to that step you will also receive some sample things to look for to prepare. If you are selected they will then do a more detailed physical and drug screen. Assuming you make it pass all that you will receive your ticket to Wilmington for the 8 week (average) training course. This is a lot of classroom and practical application of that classroom training to familiarize you with working on the railroad. This will be your first experience of some of the actual physical and mental demands of the job. After that you will be sent back to your crew base for on the job training. You will be on probation working with another experienced conductor so this is really the time to perform and demonstrate what you have learned as well as clarify any concerns you have in terms of doing the job well. This is the real test as you will be assisting real passengers on a real revenue train. After a few months (generally 6+), if you do well and pass everything you will "mark up" and start the job as an Assistant Passenger Conductor at the regular starting rate of pay (up to this point you will still be at training wages). You will work the extra board and be required to report in within 2 hours of getting called in. You will have one day off a week but it is not guaranteed (all depends on if you work an outbound trip on the day before your day off... if so you are working your day off back although you still will receive some time off after that). Most likely you will not be home every night depending on what trips you are assigned. Many are outbound one day and return the next day.
You will be expected to take the Passenger Conductor test to "promote" to a Passenger Conductor within a year or so of hire. If you are unable to pass the training and tests to become a full fledged conductor within the acceptable amount of time and tries you will no longer be employed by Amtrak. As was said in my hiring class... "Amtrak no longer hires Assistant Passenger Conductors... we hire Passenger Conductor Trainees." It is the expectation (and requirement) that you promote to Conductor. Once you promote you are back to the bottom of the extra board.
With that said, you still may work Assistant Conductor Assignments... you will just be qualified to work as a Passenger Conductor as well and sign as being responsible for the train.
There are some exceptions to what is listed above but it is mainly to illustrate that it is not an easy process. There will be lots of training, many days away from home, some nights with little sleep (just the mandatory rest period), long working days, among other situations with passengers and operations as well as emergency situations. It isn't a job for just anyone, however those who I know find it very fulfilling and they enjoy what they do.
If you are working on the Northeast Corridor some elements are slightly different and the seniority can be a bit different as well so most likely you'd stay on routes near the Corridor area. I'm from the west though so I can't fully comment on that aspect of it.
Full Disclosure: The above is the process as I went through it and as it was explained to me as well. It may have changed and there may be parts that are not necessarily accurate anymore. I ultimately went through the hiring process, however having worked in the transportation industry before and now with a young one growing up I didn't want to miss more of those "moments" per say. In the end I chose a different position elsewhere that worked better for my personal situation at the time, however I was still glad I went through the process of the hiring as it made me really think if the position were for me. Maybe a few years back before family ties and settling down it would have been perfect... but it is something to think about as you mention you have pets. Just think about how the demands of the job would fit with your personal situation and go from there. As stated above, the hours and days will be irregular for your first few years. You will hear the expression you are "owned" by the railway... that was a common expression at the crew base I was looking at. But again, it can be a very rewarding position as well. Whatever you decide... best of luck!