Thermoelectric cooled Avalanche Photodiode sensor PCB
Apparently you intend to put your TEC controller on the 5 cm x 5 cm board as your APD, and mount the board on the TEC. This strikes me as a very bad idea. Since the TEC controller works as a switcher, you are placing your very sensitive APD in close proximity to a multi-amp switching regulator. I suppose this is possible, but given what seems to be your level of expertise I would not predict good results. That said,

1) Using vias as thermal paths to reduce thermal resistance from the TEC to the APD is a really good idea. The same should be done with the thermal path to your thermistor. You do realize, I hope, that all your components, including the APD, will need to be SMTs. The amount of noise reduction you will get by using a single-stage TEC is unlikely to improve your noise performance significantly.

2) Placing your thermistor on the APD ground trace ought to work just fine for the sort of precision you want. You do realize that the thermal connection to the thermistor can be entirely independent of the electrical connection, right?

2) (the second 2) I really don't see how much you hope to gain by going double-sided on components. You'll be using a 5 cm x 5 cm board on a 4 cm x 4 cm TEC, which only leaves a 5 mm wide perimeter area. In direct answer to your question, you should get no particular harm or benefit, as long as you make sure the components are rated for the lower temperature they will experience. If you put a thermal insulator around the top of your pc board, with just a hole for the APD to peek through, you might get 50 C of cooling, which would bring everything down to about -10 or -20 C.

Another problem I see with your approach is exactly how you expect to interface the pc board to the TEC. Unless you epoxy it in place, you will need to provide pressure areas to allow clamping the board down, and you will need to do this to ensure good thermal contact. This will reduce the area available for components, which you say is already limited.