The accepted answer covers the key question, but just to answer the two subsidiary ones as well:
- Was the LM able to communicate without humans in it?
Yes. See this series of comments in the Apollo 11 flight journal, starting about 45 minutes after LM separation:
130:57:08 Duke: Hello, Apollo 11. Houston. Coming up to about 4:30 LOS. You're looking great on all your systems. Eagle is purring right along on a - After an hour and thirty [minutes] on, without any cooling, the PNGS is still looking good. Over.
PAO [about 132:00]: One of the things that we're watching with interest here in Mission Control is the Lunar Module. At the time Aldrin and Armstrong left the LM and came back into the Command and Service Module, they switched over to the secondary coolant loop as part of a test that we are running to determine just how long the Primary Guidance and Navigation System will last without cooling. The primary guidance system is not cooled on the secondary loop which provides backup cooling for the secondary guidance system and it's always been a point of interest just how long the primary guidance system will survive without the cooling - estimates range up to several hours.
PAO [about 133:45]: We presently show the Lunar Module in an orbit 63.3 by 56 nautical miles [117.2 by 103.7 km]. The Command and Service Module is in an orbit 63.2 by 53.9 [117.0 by 99.8 km]. At the time of the Trans-Earth Injection burn we expect that the Command Module will be about 1 mile below the LM and about 20 miles in front of it.
133:54:03 Duke: And, 11, Houston. For your information, Eagle, we had an ISS fail light came on at about 3:19 due to a CDU overheating. And at about this time, at AOS, it looks like we're about to lose the platform.
PAO [about 134:00]: A few minutes ago you heard Charlie Duke advise the crew that we show the LM platform is about to go. We'd mentioned that prior to leaving the Lunar Module, the crew had deactivated the primary coolant loop, placing the secondary coolant loop in operation aboard the LM. This means that the primary guidance system does not get cooling. This is part of a preplanned test to find out just how long the primary guidance system would continue to function without cooling. We had predicted in advance that it would - may not - it would possibly not last much longer than an hour or so, but at the time Charlie Duke reported the platform had just gone down on the LM, it had been in operation without cooling a little more than four hours. We had an approximate time from the crew of transferring to the secondary coolant loop of about 129 hours, 40 minutes; and we got the report from Duke that the LM guidance platform was no longer usuable at 133 hours, 55 minutes.
So it looks like there was telemetry for about 3h 45m after LM separation, at which point the guidance system failed and signal was lost.
The communication could be two-way, though it's not clear if it was for Apollo 11: the later flights sent the signal to fire the deorbit RCS from the ground, rather than from the CSM or having it programmed on a timer.
- If the above point is true, how come that no one in NASA thought it would be a good idea to check what happened with the ascent stage closer to the time, when it was probably still operational?
From the above, it looks like they kept contact until signal was lost due to the guidance system failing, and had a reasonably firm idea of the orbit at that point.
In addition, from the Apollo 11 mission report (p 103):
At lunar module jettison, the two ascent batteries had delivered 336 A-h of a nominal total capacity of 592 A-h. The ascent batteries continued to supply power for a total of 680 A-h at 28 V dc or higher.
This suggests that even if the guidance system had remained functional, the LM probably would have lost power very soon afterwards anyway - by the time this happened, the batteries were already past their nominal capacity. So it wouldn't have been possible to maintain the connection for much longer even in an ideal situation.
Apollo 9 and 10 had tried similar experiments. Apollo 9 had a slightly different rating for capacity (620 A-h) - not clear if the actual batteries were the same - and at the time of loss of signal, had obtained a total of 656 A-h from the two batteries. (Mission Report, page 9-19)
Apollo 10 let the ascent stage batteries discharge as well; it found that they gave 676 A-h out of a rated capacity of 592 A-h. (Mission Report, page 8-2). So both were fairly close to the Apollo 11 level, suggesting that it wouldn't have lasted much longer.
The problem for long-term tracking is that these three and a half hours of data only represented a couple of orbits - so it's hard to extrapolate very much from that data.