If teething doesn't seem to bother (or hasn't in the past), personally I wouldn't give meds. Some babies handle pain better than others and I've never really had to give Isabella anything for teething & whereas with Jasmine, with every tooth she had fever & I gave her baby panadol.
I think her nightwakings are more from the sch. To get to 9am when she is waking at 5.30am is a very long stretch & really depends on the type of baby you have. I would firstly put her to bed for the night a little earlier (6.30 the latest to be asleep by) so that when she wakes she is better rested and you are able to stretch her. I have been able to do it in 1 go - HARD hard work though. I had to basically take her out of the house, provide her lots of stimulation and then come close to 8.30, I would start the windown. For about 20-30 mins I was in her room (dark), holding her and rubbing her back and then in the cot at 9am. She would go down in 1-2 mins (she was exhausted by then) and she would sleep well.
But if you aren't comfortable stretching her so much in one go, I would do it in increments. So at the moment her nap is at 8am, I would add 10-15 mins to her morning A time every 3 days until you get to 9am. At the same time, push the pm nap and catnap.
Don't give up on the solids. What type of solids have you tried so far? Start with bland & soft foods like bananas or maybe avacado.... if she's not taking it from the spoon, try feeding her with your finger. That sounds yuk! but I've done it. I used to also give Jasmine a spoon to play with and feed her with another. Please have a look at the solids forum I posted a link to before. If you want to start a new post and I am sure you'll get plenty of advise on what to do....
I can't imagine how hard it is to be a single working mum & having all these nightwakings to deal with as well. We'll get through this.... hang in there.