In Azerbaijan, how do businesses' hours of operation change over Nowruz?
As a local, I can tell that in Azerbaijan, businesses' hours of operation don't change over Nowruz. Although Nowruz is an official public holiday in Azerbaijan (probably the most important one) and 20-24 March are considered non-working days, they are actually only applied to government-controlled instances and large companies.
Generally, none of the listed businesses (coffee shops, tea houses, restaurants, stores) will have different/reduced working hours during Nowruz, neither the public transport. Even government-controlled libraries and museums in Baku should work on daily basis during Nowruz as reported by this article from Azadliq Radiosu (part of Radio Free Europe/Radio Liberty):
[Novruz] Bayramı günlərində Bakı Şəhər Mədəniyyət və Turizm İdarəsinin
şəbəkəsinə daxil olan bütün kitabxana və muzeylər fəaliyyət göstərəcək.
My translation from Azerbaijani:
During [Nowruz] holidays, all the museums and libraries under
jurisdiction of Baku City Culture and Tourism Department will operate
on daily basis.
How hard will it be to find a decent room in a guesthouse or small hotel at the last minute?
Regarding the second part of the question, although some Iranian neighbors do visit Azerbaijan during Nowruz, Nowruz is a family holiday (be it in Azerbaijan or in Iran) and mostly is celebrated at home or by visiting relatives in one's home region. Which means locals travelling within country don't really use any hotels or guesthouses, they just stay with their relatives.
In addition, unfortunately, Nowruz is not branded very well in the world to attract a lot of tourists, so finding an accommodation at the last minute shouldn't be harder than at any time during the year.
How crowded will trains/buses in and out of the country be?
The last part of the original question is the only one that one should worry about. As people travel a lot to their home regions from Baku during Nowruz, trains and especially buses will be packed on regional routes (not on foreign routes though, as not too many tourists travel in and out of the country during Nowruz).
Traveling by bus from Baku:
I would recommend you to first check prices at Baku International Bus Terminal (by selecting "Tariffs" from the tabs). You will see that prices are extremely low.
Then click the "Race Schedules" tab, choose your destination and check the times.
Then you have 2 options:
Buying the tickets online by clicking the "ONLINE Ticket Sales". However, last time when I bought it online in 2014, it took me an hour to convince the driver that online tickets do exist (normally, they expect you to pay directly to them by cash). De jure, obviously, you should be able to hop in to your bus just before departure, but bear in mind that if a crazy driver starts to argue with you, it will be really hard for you to convince them that you're right with a lack of local language. However, I hope that things have changed since.
Buying tickets directly from the driver. During normal days, there will be most probably a seat for you even if you arrive just before the bus's departure. During Nowruz (or other holidays), however, you should arrive at terminal 1-1,5 hours before departure to be able to buy directly from the driver.
You should be able to come back to Baku by asking your driver when they are coming back (or the next busses). The return tickets are for the same price.
Traveling by train from Baku:
Less packed during Nowruz, because locals travel more by bus. Check the amazing online ticket search of Baku Central Railway Station. Its website has much better look than bus station's and buying online is safe.
Novruz bayramınız mübarək! - Happy Nowruz!