Point 1 doesn't appear to be true, at least looking at homicide.
"mental illness" is a category as broad as the ocean but let's try...
First let's look at psychotic patients.
Rates of Homicide During the First Episode of Psychosis and After Treatment: A Systematic Review and Meta-analysis
The observation that almost half of the homicides committed by people with a psychotic illness occur before initial treatment suggests an increased risk of homicide during the first episode of psychosis. The aim of this study was to estimate the rates of homicide during the first episode of psychosis and after treatment. A systematic search located 10 studies that reported details of all the homicide offenders with a psychotic illness within a known population during a specified period and reported the number of people who had received treatment prior to the offense. Meta-analysis of these studies showed that 38.5% (95% confidence interval [CI] = 31.1%–46.5%) of homicides occurred during the first episode of psychosis, prior to initial treatment. Homicides during first-episode psychosis occurred at a rate of 1.59 homicides per 1000 (95% CI = 1.06–2.40), equivalent to 1 in 629 presentations. The annual rate of homicide after treatment for psychosis was 0.11 homicides per 1000 patients (95% CI = 0.07–0.16), equivalent to 1 homicide in 9090 patients with schizophrenia per year.
So there's a 1 in 629 chance that someone having their first psychotic episode will kill somebody.
After that there's a 1 in 9090 chance per year that they will kill somebody.
But this doesn't give us an indicator of how it compares to the chances of being a victim.
This study may give us a better comparison:
Patients with mental illness as victims of homicide: a national consecutive case series
1496 victims of confirmed homicide died between Jan 1, 2003, and Dec 31, 2005, in England and Wales. Patients with mental illness were more likely to die by homicide than were people in the general population (incidence rate ratio 2·6, 95% CI 1·9–3·4). 90 homicide victims (6%) had contact with mental health services in the 12 months before their death. 213 patients with mental illness were convicted of homicide in the same 3 year period. 29 of 90 patient victims were killed by another patient with mental illness. In 23 of these 29 cases, the victim and perpetrator were known to each other, and in 21 of these cases, the victims and perpetrators were undergoing treatment at the same National Health Service Trust. In these 29 cases in which patient victims were killed by another patient with mental illness, alcohol and drug misuse (19 victims [66%], 27 perpetrators [93%]) and previous violence (7 victims [24%], 7 perpetrators [24%]) were common in both victims and, particularly, perpetrators. In seven of the 29 cases in which the victim was killed by another patient with mental illness, both victim and perpetrator were diagnosed with schizophrenia.
So in this sample sampling from Patients with mental illness, 90 were victims of homicide while 213 patients were convicted of homicide.
29 of 90 patient victims were killed by another patient with mental illness.
So while mental health patients are significantly more likely to be murdered ( 2.6X ) it appears that a significant fraction of that risk comes from the other mental health patients in the same trust. The average mental health patient is more likely to commit homicide than be a victim of homicide.
Claim 2 is easily confirmed:
people with severe mental illnesses are more likely to be victims, rather than perpetrators, of violent crime.