> Do we have any records of meteor impacts on the moon? Is there any way to monitor this?

Yes, and we do monitor this. There may be a deeper record (perhaps others will address that aspect), but NASA runs a lunar impact monitoring program. The basic strategy is to look for "flashes" using specially designed telescopes, where the flashes are a portion of the kinetic energy of the impact converted to visible light. NASA has been running this program since 2006. This page provides some recent candidate impacts and there's a map and links at the bottom that give a more complete accounting of what they've monitored. From a quick glance, you can see that since they started the monitoring program, they've observed ~440 candidate impact events.

> Is there any way we could find out how old the impacts are? Carbon dating?

Yes, but radiocarbon dating is (1) restricted to samples about ~50,000 years old and (2) dates the time that a living thing stopped being in equilibrium with the atmosphere, i.e., it died. Thus, radiocarbon is really not useful for the question (or for the Moon more broadly) since most craters will be much older, there's nothing alive on the Moon, and there's effectively no atmosphere on the Moon. There are however a range of radiometric dating techniques which are more applicable for the moon and impacts more specifically. In terms of dating impacts directly via a radiometric technique, the basic idea is to try to date a sample of "impact glass", i.e., material that was melted and quenched rapidly during the impact process, and thus dating this glass constrains the time of the impact that caused the melting. The common radiometric techniques applied to impact melts are Ar/Ar and a somewhat niche version of Pb-Pb ages (e.g., Zellner, 2019).

In addition to radiometric techniques applied to melts, there have been a variety of other methods proposed to approximately date lunar impacts. For example Ghent et al., 2012 suggested that the breakdown of the ejecta blanket, i.e., the blocks of the lunar surface that are excavated during impacts and strewn around the crater, could provide an estimate of age. As discussed by Ghent, large intact rock chunks within the ejecta from larger impacts are degraded by impacts of mircometeorites (there are a few other processes that also weather material on the moon which might also contribute to some degree) so the degree of preservation of the intact rocks in the ejecta can serve as a proxy for age for younger craters (once all the rocks are degraded, this method no longer works), but where "younger" is used in a geologic context, i.e., it works on craters 10s to 100s of millions of years old, which given the ~4.6 billion year history of the Moon, counts as young.