You are asking a whole lot of questions here. I'll try and give a brief answer to them:
What kind of jobs do illegal immigrants do in USA?
The majority of illegal immigrants work in low-skilled jobs:
Nationwide, unauthorized immigrants are clustered in a few occupations, notably farming, fishing and forestry (26 percent of the workforce), building and grounds (17 percent), and construction and mining (14 percent). They comprise 24 percent of all groundskeepers, 23 percent of domestic workers and 20 percent of those in clothing manufacture.
In addition, they have carved out niches in certain relatively well-paid construction trades. They hold 34 percent of all jobs in drywall installation, 27 percent in roofing and 24 percent in painting. Passel also noted that many illegal immigrants who overstayed temporary visas have higher education levels that enable them to work in office or technical jobs.
Is it worth for them to cross the border with all the implications?
In that they are doing it, the answer would appear to be 'yes'. Keep in mind that many of these immigrants are coming from very rough situations in Central America. So rough, that risking their lives to make it across the border is seen as worth it.
Do they affect the job opportunities for the American citizens?
This is a hotly debated topic.
If they don't, could a job program similar to the one used by the Eisenhower administration improve the current situation?
What is the current situation you see is the issue?