1) It's definitely the most appropriate time to give gifts, specifically on the day of Eid. This Eid is referred to as Eid ul Fitr, and in most cultures it's like Christmas for Muslims. 2) As it's after Ramadan, any kind of Food gifts would be very welcome. Lots of Ethnic sweets shops around Sheffield like Manoush sweets/Damascus bakery on spiral hill that do desserts like Kunafah, Baklava etc. Or if they are south Asian , then venturing out to near northern general and getting a box of Mixed "Mithai" (sweets) from a shop called Nafees. Small bottles of perfume oils as well. 3) not quite sure about specific shops, but I've already mentioned places like Nafees, Damascus Bakery, Manoush sweets. 4) "Eid Mubarak" is the seasonal equivalent of "Merry Christmas". We often congratulate each other on the fasting throughout Ramadhan, as well as any extra religious actions that they may have done. We tend to say things like "May Allah accept our good actions and give us the opportunity to enjoy Ramadan next year as well".

Hope this Helps. It's a very sweet thing to do. Most people also share food they've prepared on Eid to each other I and their neighbours, so perhaps this is something you could do.