This article describes how Earth's Moon helps Earth's core remain liquid, thus maintaining our magnetic field. The magnetic field, in turn, protects life on Earth from radiation. To what extent is it necessary for a planet to have a moon such as ours that is capable of helping perpetuate a liquid core to the degree that life can flourish? In particular, when we're evaluating whether newly discovered planets can harbor life, do we need to try and find those with a similar moon?
How possible is it to have a habitable planet with no moon like Earth has?
Wikipedia has a webpage on radioresistance.
MythBusters has their video: Cockroaches - Outliving a nuclear blast but the flour beetle can survive a higher dosage. Wikipedia's webpage on Tardigrades (also known colloquially as water bears, or moss piglets) has this information:
The genome of Ramazzottius varieornatus, one of the most stress-tolerant species of Tardigrades, was sequenced by a team of researchers from the University of Tokyo in 2015. Analysis revealed less than 1.2% of its genes were the result of horizontal gene transfer. They also found evidence of a loss of gene pathways that are known to promote damage due to stress. This study also found a high expression of novel Tardigrade-unique proteins, including Damage suppressor (Dsup), which was shown to protect against DNA damage from X-ray radiation. The same team applied the Dsup protein to human cultured cells and found that it suppressed X-ray damage to the human cells by ~40%.
If you like the idea of genetically modified humans (or extraterrestrials) it is possible to develop lifeforms with considerable resistance to radiation damage.
Wikipedia's webpage on Rubrobacter is incomplete but their article on Thermococcus gammatolerans (a member of the Thermococcaceae family, sole family of the Thermococci (called "Protoarchaea" by Cavalier-Smith), a class in the phylum Euryarchaeota of Archaea) says:
The resistance to ionizing radiation of T. gammatolerans is enormous. While a dose of 5 Gy is sufficient to kill a human, and a dose of 60 Gy is able to kill all cells in a colony of E. coli, Thermococcus gammatolerans can withstand doses of up to 30,000 Gy, and an instantaneous dose of up to 5,000 Gy with no loss of viability. It is the organism with the strongest known resistance to radiation, supporting a radiation of gamma rays from 30,000 gray (Gy). Unlike other organisms, cell survival in Thermococcus gammatolerans is not altered by changing conditions in its growth phase, but the lack of ideal conditions and nutrients decreases its radioresistance. The system of chromosomal DNA repair shows that cells in stationary phase of growth reconstitute DNA more rapidly than cells in exponential growth phase. T. gammatolerans can slowly or quickly rebuild damaged chromosomes without loss of viability.
The red-coloured bacterium Deinococcus radiodurans can resist 1.5 million rads of gamma radiation, about 3,000 times the amount that would kill a human. The bacterium survives and reproduces in environments that would be lethal for any other organism and it also resists high doses of ultraviolet radiation. The most important component of this radiation resistance is the ability of the bacteria to repair damage to its chromosomal DNA.
See also the Genome News Network's article: "The World’s Toughest Bacterium".
The U.S. National Institutes of Health's National Library of Medicine has an article: "Analysis of Deinococcus radiodurans's Transcriptional Response to Ionizing Radiation and Desiccation Reveals Novel Proteins That Contribute to Extreme Radioresistance" which explains:
A 5-kGy dose (5 kilogray = 500,000 rad) causes massive DNA damage, cleaving the genome of every D. radiodurans cell into multiple, subgenomic fragments (Battista et al. 1999). For most species, this level of DNA damage is lethal, but D. radiodurans has the capacity to reform its genome from these fragments in what appears to be an error-free process. The biochemical details of D. radiodurans's ionizing radiation resistance are poorly understood, but it is clear that proteins needed for cell survival are synthesized in cultures exposed to ionizing radiation.
"All of the dosimeters worn by the workers were over-exposed and did not allow an estimate of the doses received. However, information is available on the doses received by the 237 persons who were placed in hospitals and diagnosed as suffering from acute radiation syndrome. Using biological dosimetry, it was estimated that 41 of these patients received whole-body doses from external irradiation in the range 1-2 Sv, that 50 received doses between 2 and 4 Sv, that 22 received between 4 and 6 Sv, and that the remaining 21 received doses between 6 and 16 Sv. In addition, it was estimated from thyroid measurements that the thyroid dose from inhalation would range up to about 20 Gy, with 173 individuals in the 0-1.2 Gy range and seven workers with thyroid doses greater than 11 Gy (UN88). Internal exposure of those workers was mainly due to 131I and shorter-lived radioiodines, the median value of the ratio of the internal thyroid dose to the external effective dose was estimated to be 0.3 Gy per Sv. The doses resulting from intakes of other radionuclides was estimated to about 30 mSv for the early months following the accident and 85 mSv for committed dose (UN00)."
In particular, when we're evaluating whether newly discovered planets can harbor life, do we need to try and find those with a similar moon?
No. Humans may not do well with higher levels of radiation but other lifeforms can be more tolerant.