To nip these comments in the bud: asylum seekers do not need to seek asylum in the first safe country they reach. They haven't needed to do that since the 1951 Refugee Convention.
If you were seeking asylum, wouldn't you want to go somewhere you might already have a relative or friend? If there was a European war, would you be happy for the population of France to come to the UK or would you expect some to go elsewhere? That said, a lot of asylum seekers do go to a neighbouring country. Turkey currently has the most. 38% of asylum seekers are in only five countries (the UK is not one of them).
It's important to note the difference between asylum seekers, displaced people, refugees, illegal immigrants and immigrants. I think a lot of people complain about immigrants, when what they actually have a problem with is illegal immigrants.
Over recent years, the number of people moving to the UK from overseas has been higher than the number who leave, but foreign-born people still only make up 9% of the UK population.
Are they taking our jobs? No, they are not "our" jobs to take. The best person for the job will get it, if that is someone moving from overseas then they bring something UK candidates cannot offer. You could argue that successive governments have failed the UK education system in that sense.
Are they taking medical attention? Again, since they only make up 9% of the population, no. I have to wait two weeks for a GP appointment, I waited 18 months for an operation. That, again, is the government historically and persistently underfunding the NHS. It's not because immigrants are clogging the system.
Are they taking our houses? There is a problem with foreign buyers in the housing market, but these tend to be non-UK resident buyers of expensive properties.
Where is the burden on resources and facilities coming from? The UK population as a whole has increased. We are healthier and wealthier, our children survive infancy and we live longer. In the old days babies died often, children too, average life expectancy was lower. Then the world wars killed a significant number. But then penicillin was introduced, water was cleaned to drink, the NHS was developed.
In 1950 the population was just over 50 million and its increased by around a million every 3-4 years averagely. Foreign migrants make up a tiny fraction of that total number.
To answer your first question, "are people who want zero immigration racist?" Some of those people, yes, will have racist motives. We saw clear demonstrations of those race and ethnicity prejudices around the brexit referendum in some people.
Others may not be racist but may not understand that the burden on our resources, infrastructures, services, is from ourselves, not from immigrants.
As a country, we could not manage on zero immigration, it's just not feasible.