"Not allowing medical experiments on people awaiting execution."
If it's immoral to allow medical experiments on people without their consent, then it's always immoral. So not illogical.
"Rescuing dogs and whales but not caring about other animals, in fact eating them."
Eating meat does not mean one doesn't care about animals. The 'logic' of the argument is false.
"Not allowing people who are going to die soon anyway to take the euthanasia option."
I wouldn't call this moral. The law is based on religious conceptions of the sacredness of life. Laws on this are beginning to change because many people think it is wrong to let someone continue to suffer when they (the sufferer) would choose to end the suffering by getting help to die. See Dignity in Dying and similar organisations.
"Not executing murderers, instead spending a fortune keeping them in prison for life."
You might execute the wrong person. It has happened. Is execution 'moral' anyway? I wouldn't have said so, even if it's desirable in the sense of ridding society of burdens it can do without.
So, in all these examples, either the logic or the so-called morality is faulty.
Ergo, it's not illogical morality. Nor is it logical immorality.