You don't want a 36XX you want at least a 38XX. There isn't really enough memory in the 36XX series and it has no internal expansion slots.
Wifi is not easy as you will need a PCMCIA jacket for the 36, 38 and 39 series. There is a SD card slot in both the 38 and 39 but I don't know if it will support SD wifi or not.
The 3870 and 3970 both have bluetooth if this is something you may use otherwise go for a 3850 or 3950. The 39 series do tend to be significantly more money than the 38 series but for what you want there will be no difference except that the 38XX is better supported with Linux.
There is a documented problem with some 38XX in that the backlight doesn't always work when on battery. This tends not to show itself with Windo$e on but can be a problem in Linux. The HP website has the serial numbers of the faulty batch and if you have one of the faulty ones they will sort it free but I don't think their list is comprehensive and I have seen several with the fault that are not covered by HP. I also think this fault goes deeper than just the backlight as the 38XX series can sometimes lock when removed from charge.
It will run fine 24/7 and if you are always having it powered then you can buy one with a faulty battery, that makes them very cheap. If it is always powered you wil not see the backlight problem.
SSH works fine on both the 38 and 39.
You shouldn't be paying any more than about $70 for a 38XX series needing a battery, the 39XX in the same state will cost you more than $100.
Is a 5500 not a better, cheaper choice?
If you were in the UK I could sell you mine