Yes. First your employer would first need to file the proper petition for whichever category you're applying for. The process is different for each category. Once your employer's Petition is approved, you file I-539 Change/Extend Non Immigrant Visa. When I-539 is approved, then you are in either H1 or J1 Status. The approval could take from 6-12 months.
Yes, you are not qualified for an H-1B and since you are already on a training status, I doubt that the USCIS would approve you for a J-1 training program.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
You do not appear eligible for an H-1B non-immigrant visa. However, please keep in mind it is difficult to assess eligibility for a visa with the limited information on a public forum. You may be eligible for a J-1 non-immigrant visa. The two-year home residency requirement is attached to some J-1 non-immigrant visas depending upon whether your career is on a specified skill list. This requires you till reside in your native country for two years before seeking future immigration benefits. There is a limited waiver that you may be eligible for down the road.
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An H1B or H1B1 requires the beneficiary to be offered employment in a specialty occupation and the beneficiary to possess at least the equivalent of a 4 year U.S. bachelor's degree. The J1 alternative in your email may be successful if the employer issues you Form DS-2019.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter, nor should it be viewed as establishing an attorney client relationship of any kind.