You can explore the small shops, shacks and stalls in Meena Bazaar in the old part of Dubai and find haberdashery, fashion accessories, textiles, jewellery, footware and fabulous ethnic food.
No you absolutely cannot drink alcohol in a mall, BUT many malls have hotels connected to them, such as The Address Hotel in Dubai Mall where you can walk directly to the mall. You can consume alcoholic beverages in a hotel connected to a mall. Not a good idea to wander around a mall if you’ve had a fair bit to drink though!
Head towards the Kite Beach area in Jumeirah 3. There a 5km or so green walking/running track along the sea here. It’s completely free to access and you pass a lively, vibrant set of cafes and restaurants, the fishing harbour where you can buy fresh fish from the Fish Market and if you keep walking you get to a wilder, quieter part where you can go stand up paddle boarding and walk all the way to the canal. This section is near Sunset Mall on Jumeirah Beach Road. 
The weather in Dubai seems to be changing along with the world’s weather pattern with a lot more rain than we’ve seen in recent years! Months to AVOID due to it being incredibly hot and humid are definitely June, July, August and September. The end of October and most of November SHOULD be safe, sunny months to visit with the humidity being tolerable. Most of April should also be a safe bet. Try and avoid UK half term breaks as things get expensive and very busy in hotels.
Yes absolutely there are many studios offering high quality yoga classes and even teacher training 200 hour certifcation classes in Dubai. Always ask to do a trial class first, often these will be free and try and establish the level and range of classes and teaching suits you. You could also opt to not join one studio but do different classes at different places using apps like Class Pass. I like Inspire and Studio Republik but make sure you can easily get to your chosen classes as else you won’t go! We have done a post on some yoga classes and teachers in Dubai. A good starting point. 
It’s quite big and growing all the time. There are more more vegan restaurants in Dubai and of course more vegan options in on vegan restaurants. Remember alot of South Indian vegetarian food is naturally vegan and very good in Dubai. Try Kamat in Business Bay or Bikaner Wala in Al Barsha. Or you could always try a and make your own vegan food using one of our vegan recipes quite easily at home. 