Thanks for your patience.
They will not receive a UK passport just because they have married you.
To receive one, they would first have to apply for and obtain a spouse visa to live and work in the UK with you. The webpage on this is here:
Only once she has received the spouse visa would she be able to work in the UK.
She would then have to live in the UK with you for 5 years, extending the visa once and then apply for indefinite leave to remain. Indefinite leave to mean is the right to stay in the UK without applying for any more visas and independently of their marriage to you.
Once she has obtained indefinite leave to remain, she could apply to naturalise as a UK citizen by applying for naturalisation provided she still meets the requirements:
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No. She would be not be regarded as an EU citizen for the purpose of entering other countries.
Whether she would need a visa to enter those countries would depend on the particular country's visa and immigration rules.
If you were to work periodically outside of the UK then it would be okay for her to stay in the UK but you will have ot demonstrate that you remain in a loving relationship together.
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