Do most people speak English?
I don't speak Arabic, but got away with doing day-to-day things in English. With less than 20% of its population of its 8 million population being Emirati, most workers in the UAE, especially those in the service industry, seem to be from India, Pakistan, the Philippines, other Arab countries, or other expats. Even though not everyone speaks good English, but there shouldn't be a huge problem communicating for the most part. However, at times it could be a bit frustrating if you want something specific and couldn't get your exact meaning across.
Do you have to wear an abaya?
You don't have to, but it's not a good idea to walk around in a tank top with shorts either, as you may get some unwanted attention. I got an abaya made with stretchy fabric for around 30 USD and usually wore shorts and whatever I wanted underneath. Occasionally I wore a head scarf, mostly for protection against the sun. Most of the time, when I wasn't in an abaya, I just wore thin long-sleeve shirts and long pants with sandals. This might sound obvious, but avoid see-through materials or men might feel entitled to make some comments about how you look (it happened a few times when I thought I was dressed modestly enough). The general rule of thumb is, if you're not sure if your outfit is appropriate, it probably isn't. Better safe than sorry! However, I don't think anyone expects a foreigner to be fully covered or to wear an abaya all the time.
Is it safe for women?
While I can only speak from my own experience, I felt pretty safe normally despite feeling uncomfortable and paranoid at times. I walked around and took taxis alone during the day and at night without problems. However, I have heard horrible stories of women being raped and killed so maybe I was just lucky. Occasionally, men stare at you quite openly or sometimes even follow your around. Sometimes, even if you're just acting in a friendly fashion, it might be misunderstood as you being interested so be aware.
One time a taxi driver did make some unsavory comments about his manhood (he gestured that it was the size of his forearm... hrm, right) and invited me to be his third wife in Pakistan. That was when I rolled down the window and repeatedly yelled, "Haram!" -- which was one of the very few Arabic words I learned meaning forbidden/sinful. The driver then said he was just kidding and followed my request to let me off.
There are loads of nice hotels, but I find that ones in Dubai tend to be a better value. I stayed at these 2 hotels while visiting Abu Dhabi and found both to be a decent choice both price and condition-wise. I booked through booking.com and both were under 100 USD per night. I liked Al Rawda Arjaan by Rotana much better than Mercure Centre Hotel because it's right next to Al Wahda Mall mall, which is a very short walk to the bus terminal. Also, the room is much bigger.
How's the internet connection?
Most hotels have decent wifi. For your apartment, you can get internet installed. I went with etisalat and my connection was pretty fast. Some sites are blocked by the government, mostly porn or if the site you're looking for contains a keyword they block. I didn't have problems accessing most sites though. It doesn't seem to be as censored as China.
How often does it rain?
Not very often at all. I think it rained once or twice the whole time I was in the Emirates. Once in the middle of the night and another time for a really short time (for 30 seconds while I walked from the taxi to the mall).