Puppy peeing and pooping behaviours!

To your second question:
That will not magically happen. She knows that she is allowed to use the bathroom inside on these two spots, so she will continue to do so. She doesn't know and understand that you actually want her to go only go outside.
If she goes outside, are you praising her, treating her, throwing a puppy party for her?
To 3: I don't believe that there is a natural house break habit. So yes, keep going outside with her, but as often as necessary. So, whenever you think she needs to go (shortly after food, after a nap, after play/training, basically every 30 minutes to an hour) take her outside, it's about her digestive schedule right now, not your set times for going outside (that comes later when she learns to hold it).
Are you living in a house with a yard or apartment building? If a house with a yard, just go in the yard with her, let her explore, but the novelty will wear off quickly. If you then praise her, maybe even give her a cue for going potty, she will get in no time that you want her to go to the bathroom outside. BUT take the pee pads and news paper away, don't give her the opportunity to go inside anymore, so she has a chance to understand the concept.
Important: Supervision! Don't let her out of your sight and take her outside as soon as you see signs (like sniffing around a lot) and praise, praise, praise

I went through the exact same thing recently!! My 4.5 month old JUST got the hang of housetraining (with a few accidents still but that's ok).

I started with pee pads and I really didn't want to but I work full time and I'm only able to come home during lunch hour. I basically had to learn how to housetrain her all over again when she was old enough to hold in her pee for a few hours. I removed all her pee pads cold turkey. I find that that made me more diligent to watch her and take her out to avoid messes in the house.

Do you work full time? or you home most of the time?

It was a really tiring process but these were the rules I followed:

- every 30-40 mins I took her out to pee/poop. If she did any business outside I would do a little happy dance (basically looking like an idiot) and gave her a really good treat (not kibble)
- after drinking water, took her out after 20-30 mins
- if I saw her circling I knew she had to poop so I would say "nono" in a calm voice and carry outside. Again, happy poop dance and yummy treat.
- While she is peeing and pooping I will say "go pee!" or "go poo!" and once shes finished I will praise and treat. She can pretty much pee on command now.

If she didn't pee or poop when I took her out, I would bring her back in for 5 mins and then take her back out. They key was being able to catch her doing it outside so I could praise her.

At first I had to bring a pee pad with me outside (sometimes with her pee already on it). I looked like an idiot but that was the only way to encourage her to do her business outside. Too many times I would let her out, nothing happened, we would come back in and she would pee in the hallway.

I had to watch her like a hawk. I learned to read her body language really well. Sometimes though, she just pees too quickly so I just had to silently clean up the mess and be more watchful for the next time.

My pup works really well on a schedule. She knows now that I come home every lunch hour and as soon as I take her out she will pee and poop. Same as when I get home from work.

I believe for feeding under 6 months of age you feed 3 meals/day. 6+ months it's twice and eventually you can feed even once a day. I still feed my 12 year old dog twice a day. My pup gets too gassy if the meals are too large and she will get really soft stool. Their little tummies can't handle portions that are too big so stick with 3 meals a day.

I know this is hard to believe now, but one day your pup will just "get it". That's what happened with mine. I was watching TV and she was playing on her own and she just ran to the door and sat down and looked at me. She was telling me she needed to go out! I was so happy!

Just keep at it, persistence and consistency is key. She will get it eventually!