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In the exercises of thirds instead of playing it 1/3 2/4 3/5 can I play it 1/3 2/4 1/5 or it's not advisable? It's really hard for me to raise the fourth finger when going to play 3/5; are there exercises for that part to make you easily move from 2/4 to 3/5 smoothly and legato?

I'm new to piano, so I apologize if this is a silly question.

All fingering is flexible!

BUT, in a passage of 3rds, confined to the '5-finger position' you need a pretty good reason NOT to use 1-3, 2-4, 3-5. And 'I'm a beginner, and I find it hard to lift my 4th finger' isn't a good reason!

It sounds as if you're playing an exercise that covers that problem already! Stick with it. Check with your teacher that your arm, hand and finger positions are correct.

Whatever you are playing on piano, it is important to plan out the fingering so that you get the articulation you want and are able to get to what follows easily. The exact fingering is going to be dependent on the context of the piece, in combination with what works for your hand. It is advisable to try out different fingering options to decide what is best.

Take a look here as an example - Chopin, Etude #6 gis-moll. This is just a reference, there are a couple of ways, I think Liszt had his own way. As another reference, you can take a look at Czerny's 'The Art of Finger Dexterity', op 740 #10

A very long time ago, I took piano lessons briefly.

I collect memory gadgets. The piano teacher knew this, and she gave me one: NARF. Notes, Articulation, Rhythm, Fingering. You want to think about ALL of these.

This sounds like a practice exercise, as opposed to a piece. It is designed to get you to the point that you can finger what you need, when you need it, how you need it. As such, it is worth the effort to work on it, as written, 1-3, 2-4, 3-5 and all.

Carl Philip Emanuel Bach offers always different fingersettings and if he didn’t ... I would! Because I think the purpose to be be able to play a piece in a comfortable way and enjoy playing is as important to train the independence of each finger. Not every piano player has the aim to become a professional pianist. The joy of making music should be in a positive balance with finger training.