There's no doubt Cochrane knows his astronomy at least as a hobby (he can find Leo even while totally wasted) so the telescope is definitely his. But then, being an amateur astronomer does not imply having the wherewithal to design and build a space ship capable of FTL travel.
A further thing to remember is the fact that Lily gives a lot of clues that suggest she might be ex-military. Tellingly, she tells Picard "You're not the first man to get a thrill from murdering someone! I see it all the time
!" So what the hell has Lily been doing lately that has involved seeing a lot of murders? When you consider that when Picard finds her she is shooting at him with a machinegun in a nuclear missile silo, she probably means that she saw it happen a lot during the war, which further explains how she got access to the missile silo in the first place (she was probably stationed there when the war ended).