Medieval city trading hands frequently over 200 years

I'll second the other answers about the culture. It will be a mix.

I also would like to address "the conquest" part. It was very common for some cities in ancient and medieval times to change hands and/or be sacked multiple times in a century (see Ephesus) for example. What's remarkable is that those cities didn't do much worse (in terms of economy and population) than their "peaceful" peers - providing there was no outright massacre or enslavement.

A sack of a city, while a very traumatic event for its inhabitants, is getting healed relatively quickly. Even better if a city, through political maneuvering and paying tribute, can avoid being sacked altogether. Generally, a life in old times has no security guarantee. If there is no impeding invasion, inhabitants of a city wouldn't think about going to other city just to give them more security.

There are two questions here. I'll tackle the culture question first. In parts of the world that have been under the influence of different cultures, you tend to see a blending. Language, food, art, clothing, etc would pick up influences from the different ruling powers. As a trade center, it would also have influences from its trading partners.

Now for the question about how the people would feel about facing capture. Bad. They would feel bad. If the years in your world are the same length as earth years, your people have been through some tough times. Someone who is in her seventies will remember two different conquerings. This frequent warfare would likely have a bigger cultural impact than the different ruling cultures. The fact that losing wars appears to be a fact of life could stifle economic and scientific development and make people really unhappy.

Given the amount of trade involved, the culture probably is a mix of both sides, and would have been even without the conquests. Much depends on how much the "Islamic" side imposes restrictions (the Romans were pretty cool about most stuff).

As for the conquest, that would depend on the amount of damage they expect. If the soldiers traditionally pillage the city, the inhabitants will dread it. If it's settled by the armies clashing outside the city and no one thinks the worse of you for opening the gates to the victor -- that would depend on the changes of law you expect, which would chiefly be in area of taxes and regulations on trade.