Sorry but I shall have to be brief --
1. A slightly discharged car battery (for example) will be close enough to 12V (if that is what your modem/router requires, of course). As I also mentioned, the battery pack from a power tool could be used. (I'm not sure who (without checking) but other
kitizens have successfully experimented that way.)
2. Transmission line theory is something I know very little about. At best one needs to consider the (transmission) source, the terminating load (of the receiver), the resistance, the impedance, the capacitance and the inductance of all segments that make up the circuit. Undoubtedly complex numbers, differential equations and integrals will also be part of the theoretical mix. I tend to ignore the subject in detail and just regard it as a black-box effect. It is only when someone (like
JGO) makes a mention of the subject do I realise that my simple DC circuit theory may not be giving the full picture. The (English) wikipedia
page would be a good starting point for any understanding.