1: It should never be unclear why a lender is denying your application. They have a legal obligation to inform borrowers of their credit decisions with a legitimate reason. If you're working with someone incompetent who can't tell you that reason then you need to push harder to get one.
2: Appraisals are always non-refundable. The lender isn't going to pay the appraiser, and the appraiser isn't going to take $0 for all that work just because you and the loan officer didn't do your due diligence. You can't transfer appraisals simply because of guidelines. It gets tricky and some lenders allow appraisal transfers, but most competent appraisers won't just transfer the appraisal. It's technically a new assignment even if it's the same appraiser because the client is different, the client is the lender not you. Appraisals are good for 120 days on conventional and FHA. 150 days for USDA, and 180 days for VA...not 60 days so that's just an excuse that doesn't really matter and probably given because they don't want to get into it with you about why the appraisal can't be transferred. A lender can approve whatever appraisers they want, it's a simple process so saying "they aren't approved" is just another way to avoid the conversation with you and get you to move onto the things you should actually be focusing on. One of those being that you'll have to get a new appraisal to work with a new lender.
3: The reduce the risk of have some lender deny your application...work with a better lender and better loan officer. It's a simple process to get a lot of certainty before you move forward and at that point only a very small amount of rare scenarios can prevent or delay closing because something needs to be done, and in those cases the reason will be obvious to you. This is why you need to dig into why the lender is denying your application. Get a reason and don't settle for some incomplete explanation. They ultimately have to send you a denial letter so you can also look on there.
4: Applying to 3-4 lenders is just a crap move. You can inquire with those lenders, and technically yes you can apply but it's kind of a jerk move and a huge waste of time. This is why I suggest finding a really good loan officer first so nobody is wasting time or money. A really good loan officer won't waste their time because it's very valuable and they'll be straight with you upfront.
5: I'll never pay for a client's appraisal. Most lenders will never pay for a clients appraisal. That's not a realistic thing to look for
You just need to find a really good loan officer that knows there business. It sucks that you wasted $1300 with an incompetent loan officer, but that's your main problem. Find a better point person. Ask tough questions, do your research and educate yourself.