All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Your question includes two matters:
First, the ruling of forged certificates in order to stay or to work in a non-Muslim country.
The answer to your question is that basically doing so is forbidden since whoever enters a country, which is at war with Muslims, with an agreement, has to abide by their laws as long as these laws are not contrary to Islam. Violating these laws is a breach of the agreement.
However, falsification is allowed in a non-Muslim country just as it is allowed in a Muslim country in the following two conditions:
A. If a person cannot receive his legal rights except through falsification.
B. When he is compelled to use falsification, otherwise he cannot live, for the verse (which means): {while He has explained in detail to you what He has forbidden you, excepting that to which you are compelled}[Quran 6:119].
Second, the ruling of living in a non-Muslim country, the answer to this question is that living in such a country has two situations:
First, a Muslim who is able to perform his religious duties and does not fear his faith and religion to be affected; migration for such a person is not obligatory but it is recommended.
Second, a Muslim who is incapable to manifest his religious duties in that country or he fears for his religion: in this case migration becomes obligatory and living there is forbidden.
For more details refer to our previous Fataawa: 84163 and 81642.
Allaah Knows best.