Once a cat is fully grown (around 8 months) you can't tell its age by its appearance. She could be one or ten and look the same.
Presuming the cat doesn't have an owner who is missing her, it would be great if you could provide a loving home for her.
Claws are a natural part of a cat. They have no idea that they can hurt people with them. To a cat, their claws are simply a way of grabbing things. Be patient! Don't get her declawed. Declawing amputates each finger/toe at the last digit, cruelly mutilating the cat. I believe this practice is illegal in Australia and if it is an option in your country, I advise you on behalf of your cat, not to do it.
The smaller she is, the sharper her claws will be. However, larger cats are more powerful, so can still create unintentional damage. If she will be inside most of the time, get her a scratching post. You can also get a claw trimmer that snips the sharp part of the claws off. Some cats tolerate this very well. Other cats freak out. Don't trim the claws too far back or you will hurt her and draw blood. Avoid the pink area (living tissue).
Even if you trim her claws, she will still need to scratch, whether it's outside on trees or inside on a scratching post (or your furniture, if you don't provide a scratching post!). Scratching removes the outer layer of the claw when it has become loose, revealing a fresh, sharper claw underneath. The fresh claw gradually bluntens as the cat goes about her business until another one is ready.
You will need to apply some cat psychology. Why is she climbing up your leg? One of my cats does this when she wants a cuddle (and she won't take no for an answer!) One of my mother's cats used to do this when he wanted food (human or cat!) My mother initially thought it was cute until he started hurting her on a regular basis.
I usually wear jeans, so the claws don't hurt me, but they do pull threads on my clothes, which I'd rather they didn't. You might need to wear thicker pants while trying to discourage her from climbing up you. This will take time, and she will most likely never stop doing it completely.
Cats learn how to play as kittens. Humans will either teach them to play rough (claws out, attack immediately) or gentle (claws mostly in, no attacking). The trick is not to encourage attacking behaviour, and not to sharply withdraw your hand if attacked (this takes some doing!). If you withdraw your hand, that become part of the game, and she will enjoy "winning" by catching your hand, unintentionally scratching you at the same time.
If she starts attacking you gently and you can bear it, leave your hand completely motionless and limp. This quickly becomes boring for her (she wants a reaction) and if you're lucky she will learn not to attack you in play. With all of my three cats, I can quickly reach down and rub their tummies when they're on their back without fear of attack! Some other cats would instantly attack, but this is mostly a learnt behaviour, not instinctive. The more she trusts you to do this and she enjoys it, the less likely she is to use her claws on you, even in play.
There are many treatments on the market. Some you can get from a vet, some from a pet store. While they are expensive initially, they usually have multiple doses, so the cost is spread over many months. This doesn't help if you have very little money right now, however!
Fleas prefer cats and dogs, but will bite humans if they can't find a furry host. Flea bites are painful. I know from personal experience! Fleas are remarkable little creatures with many survival strategies. If she does have fleas and you have carpet, depending on the season you might already have flea eggs in your house.
Flea eggs can survive for six months and are usually dormant over winter, so you will need to keep an eye on the situation. Vacuuming is a good way of getting rid of most of the eggs, the trouble is that you don't know where they are so you have to be very thorough. Even so, you might miss spots that fleas can get to and you can't. If she goes outside at all, you will need to keep treating her for fleas. My cats got fleas from a lost kitten that I rescued (before I found its home) and it took me two years to get them completely flea-free again.
You can get free local advice at a pet store, although a vet might give you better information (depends both on the pet store and the vet).
Liquid discharge from eyes
Eye infections are very common in cats. Some are serious, some are less so. It could clear up by itself, or it could need treatment. She needs to be examined by a vet to determine the best course of action. Some treatments are effectively free in the less serious cases, using things you already have in your house. She might need medicine. Only a vet can advise you.
You shouldn't let the cat have unsupervised access to the baby, for both of their sakes. Cats are generally good with babies, recognising that they are infants, and not fully rational. Growing up, we had the nastiest cat ever. He would not hesitate to bite you or attack you at the slightest provocation. However, he never hurt a baby, no matter how much he was provoked.
If this cat has some trauma, she might have triggers that your baby (or you) set off accidentally. It might take months before she gets triggered, or she might be mentally healthy and it's not a problem. She's more likely to get triggered as you all get more comfortable, and you accidentally do something innocuous you think she'd be fine with, but she reacts badly. Or she doesn't have any triggers. Only time will tell.
One of my cats (an ex-stray) didn't like me touching her head for the first couple of years (some unknown previous trauma), but now I have her trust and she loves it!
Make sure she's sterilised. If she's already been done, then you know she's come from someone's home, which seems likely considering how friendly you've described her. Obviously, only a vet can do this for you and you don't want her to have a litter (or multiple litters) of kittens that you have to find homes for, so it's another reason for getting her to a vet sooner rather than later. Vets in Australia won't desex a pregnant cat but your country might differ (not advised).
If things go well, she will become part of your family. If your baby was sick, would you wait until you could afford treatment to take it to the doctor? Look into ways the cat can get treatment now, before any undiagnosed issues become serious. As mentioned in the comments, I can't afford vet treatment, what are my options? might be able to help you. Long term, owning cats has been quite affordable for me. I guess food would be AU$20 per month, preventative treatments AU$10 per month, plus annual health checks and the occasional sickness (one per three cats per year).
Good luck! She sounds adorable and quite lucky to find a family who is willing to take her in. I hope she is a furry bundle of joy for you for many years to come!