The majority of immigrants are unlikely to be wealthy (and wealthy immigrants are unlikely to be affected by strict immigration restrictions), so making it more difficult for those people to move to a place that they feel gives them an opportunity at a better life is likely to make things worse for those people. People who favour social ideals are probably less inclined to differentiate between poor people here and poor people there.
The issues with a lack of jobs are at least partly because our society really doesn't need so much labour these days. We could easily meet everyone's basic needs without requiring everyone to work 40 hours a week, and still find plenty of time to produce luxury goods, but instead we continue with a system of artificial scarcity and competition. That system only gets worse as technology advances, the rich hoard their resources and the majority of people remain ignorant while fighting over scraps. I'm not going to say it'd be easy to establish a more efficient system, it's a logistical and social challenge, but it'd be worth it.
I'd probably agree that social equality would be likely to come before economic equality. We have to stop finding excuses to separate "us" from "them" and start treating everyone as people (and that includes ourselves). I'd also tend to agree that this is where training (or rather, education) comes in, we need to start moving society towards the idea of working together and treating people with respect. I'm not sure if empathy is something people are born with or if it can be learned but I do think it should be a priority to try and help people understand each other and see that they can be their own person while allowing others the same.
Sort of an aside, but I'm not really keen on using the term "equality" because I think it implies a certain "one size fits all" mentality that doesn't really hold up. I tend to prefer "fairness", although I'll admit that it's not exactly the most precise term either. Not really relevant, but I thought I'd mention it.
Socialism is cool but I'd rather pursue the career of my choice and make as good a living I possibly can that way and be rewarded where I work hard, rather than limit myself to the earning potential dictated by hard people work 'collectively'. Socialism works well when everyone would pulls their weight, but without the incentive to go further, I cannot see how people can maintain high motivation beyond doing the bare minimum required to get by.
The idea of a system that rewards hard work and contributions to society is appealing but it's unfortunate that no such system currently exists in our world. It's true that working hard can improve a person's chances of financial success but it's neither necessary nor guaranteed to help. Indeed, paying as little as possible for work to be done is seen by many as a key factor in running a profitable business. It's generally more important to be perceived as someone who is of value to an organisation/individual (and thus a source of potential profit) than it is to actually be a productive worker. That our political system is a superficial show and popularity contest is merely representative of the overall way our society currently operates.
The reality is that many people are already working hard for reasons other than personal gain, and this despite the fact that society rarely rewards such efforts. We have charities because there are needs going unmet by profit driven systems and people still wish to help meet those needs to benefit others and society as a whole. People work hard in low paid jobs because they have personal pride in their work and feel that putting in less effort would go against their personal beliefs. Some people work merely for the recognition of others and the status of achieving certain goals or positions. Many people are already doing the minimum required to get by and working in jobs that they don't like because they need money to survive, many of those people could probably be doing something more productive and helpful with their time (while also improving their own lives) if they weren't bound by the basic survival requirements that our system enforces.
Our society may be better than some alternatives but it's really very far from approaching something that's actually good.