Yes, I am bias because I'm a Realtor, but this is a perfect example of someone needing assistance. You seem to have a lot of questions, which is perfectly fine . . . and a decent Realtor would be able to answer them very quickly with no problem. Typically when you're buying a property you're getting your Realtor for no out of pocket expense, so you're basically wasting the opportunity if you do not get one (seller offers a set commission . . . listing agent would offer out half of that commission to any agent bringing a buyer. That's how the buyer's agent gets paid . . . sometimes banks don't work well with this, so talk with your realtor about their policy if a bank tries to cut out their commission).
Now to answer your questions . . . it's hard to say whether offering over the asking price would make a difference. In some cases, the bank would still be losing a ton of money even if you offered over. Your best bet, if you like the property, offer your highest and best offer up front. Most likely it still will take the same amount of time because a lot of the hold up is because banks are overwhelmed. They are not in the real estate business and most do not have procedures set up to deal with the amount of inventory they have.
#2) Yes, you should always have a Realtor, especially when dealing with a large company. The wait time on REO properties are a lot less because there is no need for paperwork and negotiations between the seller and lien holder. At this point the bank owns it, all the paperwork is done, and the bank knows exactly what they would need to get for the property. There are lot of other odd quarks that your Realtor would help you look out for though . . . for example, I can almost guarantee that the bank is not going to fix anything in the home . . . also most of the time utilities are out so it's difficult just to be able to get the home inspected (for personal reference at this point, I'm sure). You always want to have your own representation (realtor, lawyer) because the incentives that the bank gives, like closing with their settlement company for example, are not worth it and could cause problems in the end (I was told by a lawyer specifically that it is typical for the selling bank to cut corners on things like title searches, so just go ahead and pay for it on your own.).
#3) Your best bet for buying any untraditional property is to work with an experienced Realtor. They will know the trends in their market and will know the best lawyers and settlement companies to work with in these situations. There are a ton of risks with these properties, like liens left on the titles (REO properties often offer "insurable title" not "marketable and insurable title") so you want to have as much help as you can, to determine if the potential benefits are worth the risks.
Also, don't be surprised if the foreclosures and REOs are not as good of deals as you though. Since so many of these properties were purchased during the inflated markets, the amount of money needed out of the properties are way above market value.
Typical sales and new constructions can be good deals now. A lot of people, for whatever reason, are just deciding to sell their home and have come to grips with what the home is worth. Also, a lot of the national home builders have tons of inventory and can afford, more so than traditional sellers, to cut their losses and just move the properties (I've seen, on multiple occasions, prices $100,000 off of ready to move in new construction properties). You may be surprised at the deals out there . . . it truly is the best time to buy!
VA & NC Realtor.