Can Apple Tell If An iPhone Has Been Jailbroken?

I have a few problems with my iPhone 4s. Its been running on IOS 6 since it came out and has been jailbroken since December. I've had no issues with it being jailbroken being that this is the second time I've jailbroken this phone. At the beginning of May i had a teacher fix the front screen because it had been shattered. Everything as going good until i got home and noticed my phone wasn't connecting to Wi-Fi. I go to settings and it my home Wi-Fi wasn't even showing i was around Wi-Fi! I go into my mother's room where the tower is and in connects but as soon as i leave her room it goes away. So i figered if i could take it to Apple they could fix it but it being jailbroken they won't touch it. So i just connected my phone to iTunes and restored it. So a couple of questions,
Will Apple be able to fix my phone or will i have to buy a new phone?
Will Apple be able to tell that I've jailbroken my phone twice?
What is wrong with my Wi-Fi signal? It connects to my computer and Kindle but not my phone.

I think that kind of software not introduced yet, So take it to the AASP (Apple Authority Service Provider) before you taken to the apple clean restore your device first and then take it another AASP don't take it to the same Centre who won't touch your device.