1. Dear Arslana, once you have been living in the UK for five years lawfully, with both the right to live and work in the UK, you can apply for Indefinite Leave to Remain and one year later, for citizenship. So, if you had the right to live and work in the uK before you got married in July 2013, which all adds up to five years, then you can apply for Indefinite Leave to Remain. This then gives you the right to reside permanently in the UK. One year after you have got ILR, you can apply for citizenship.
2. It is not possible to simply apply for citizenship after five years living in the UK. You must be able to show you were living visa free in the uK for one year before you can make your citizenship application. However, once you have got ILR, you are no longer required to get a visa to live in the UK. You can live in the UK indefinitely.
3. The issue of the UK leaving the EU will only affect the visa status of your spouse who has an EEA passport. It is still to early to say what arrangements will be made, but it is likely that he will have to apply for a visa if he wishes to remain in the UJk. However, I would suggest that he also seek to obtain UK citizenship as this will regularise his position. The first thing he should do is apply for permanent residence card as he will also need to be able to show he was visa free for a year before he applies for citizenship. Here is a link to the UK government website which explains how to apply for a residence card. https://www.gov.uk/apply-for-a-uk-residence-card/permanent-residence-card
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5. I answered the question on the basis that you were a non EEA national, so I don't know how you believe there has been a misunderstanding. It is five years for a NON EEA National to apply for Indefinite Leave to Remain. However, for a non EEA National, time spent as a student does not count towards these necessary five years for Indefinite Leave to Remain. So, you won't be able to apply for ILR until July 2018. The advice I gave in paragraph 3 above is directed towards your spouse, the EEA national. I think this might be the source of your misunderstanding.