Please refer to a series of mas'alah starting from # 308 onwards in the Tawzihul Masail of Sayyid Sistani.
But to summarize it all, In your condition you do not need to make an extra wudhoo for your second salah, you should make one wudhoo in the times that you're mostly paak, and pray your salah(s) with that. You can certainly recite Quran after salah. It is better you don't touch the writing, but if you are in the state of wudhoo, you can touch them.
Ask an Alim Team.