Oh bless you, it is SO hard early on and it's really early days- don't write yourself off just yet!
I have a 15 month age gap between my DS and DD. DS is now 2.1 and DD 10 months. Those first couple of months were really tricky and it was tough finding my feet looking after 2 tiny ones during the day, but we survived by sticking to DS routine (and incorporating DD into it as much as possible, now they both have a post-lunch sleep at the same time), making myself get out and about as much as possible with them, even just for a walk or to playgroups, and reading/TV/ANYTHING to keep DS occupied whilst I dealt with DD. I remember bawling my eyes out the in run up to DP going back to work after paternity leave, and thinking "how am I going to cope?"- I really did feel sick. Looking back now a lot of this was baby blues and hormones. We survived, and (eventually!) thrived after being thrown in the deep end.
I was also worried about my relationship with DS when DD came along, but my experience was that my bond with DD has actually taken longer to build as I always wanted DS to still be my 'baby', so felt so guilty about him possibly feeling displaced, so I think I lavished more attention on him rather than DD. Now I love DD just as much as DS, but it has taken a little longer to get there. Thinking about it, DS was too little to be jealous and really couldn't have given a crap about DD arriving, as long as he got fed/cuddled/played with- it was me who was fretting too much!
Your DD sounds like she's being a normal toddler. DS would ask for anything fun over and over again, it's normal. There is light at the end of the tunnel- DS and DD get on great, and now DD is older, they keep each other occupied. DD adores DS and follows him everywhere, DS adores DD and tries to play with her (although hates it when she tries to nick his toys!). How I felt when DD was first born feels a world away, and hopefully it will for you too.
I think it's great that you're going to the docs now if you're worried. I didn't have PND with DS, but after having DD I think I had mild PND up until Nov last year, and wish I'd sort help earlier, as I spent quite a lot of time stressed over minor things and in tears on the phone to DP whilst he was at work, and i'm sure this could've been sorted sooner, as the kids were really not that hard work once DD reached 3-4 months.
Oh and re feeding: It's bloody hard work breastfeeding when you have a toddler too. Obviously do what's right for you and definitely carry on if that side of things if it's working, but if it's not, and your struggling to entertain DD1, and it's making you more stressed and everyone upset, then seriously consider switching to FF. I might get flamed for advising this, but (again, my experience) when I switched to FF with DD I found it so much easier, as I wasn't trying to juggle baby on boob with nipple shields/demanding toddler, and DP could help me more.
Good luck- you WILL get there!