Is building a house feasible with my budget? Or should I give up the dream?

The plot is not easy to build on,

You should explain that in more detail, as it affects the required budget.

Reducing from 1.5 stories to 2 stories is also a tip I’ll take to save money,

You misunderstood that (or summarized it incorrectly). The "Anstattvilla" is (at least as a basic model, with a slope it looks different again) the cheapest construction – only in the case of your desired size does a second straight-wall story not pay off. However ...

Exterior dimensions for the house 7x12m

... this would neither yield the desired 110 sqm nor would it be a favorable aspect ratio. That would be about 100 sqm, and a long side in the range of 1.2 to 1.6 times the gable side is usually cheaper. "Square" is not bad either – only "dogmatically square" regularly leads to the need for extra surface area (or squeezing, or to a convoluted layout). The "bestseller" Town & Country Flair 110/113, by the way, has a floor aspect ratio of 1:1.07.

If you are planning for a family, that definitely changes the situation significantly.
110m² without a basement is already really compact for a family. When you subtract the building services and circulation space, you're left with the equivalent of a 70-80m² apartment. Possibly reasonable with one child, anything more than that is rather spartan.
So more like 140m². That definitely doesn't work for 350k.
If your girlfriend supports you in all of this, she should probably do so financially as well if you want to turn the dream into reality. Otherwise, the family home won’t happen.
A quick rough calculation for you:
140m² x 2300 (lowest standard + own contribution) 322k
Additional building costs 40-50k
Sparsest outdoor facilities 10-20k
Unexpected costs and what I forgot 20k

This results in a realistic calculation of over 400k for your modest little dream, and that is still optimistic and with the simplest execution / equipment.
So I say: for 350k total costs, you currently can't build a family home in RLP. If it is really your dream, you have to adjust the volume, be it equity from the family or income from the girlfriend.
Or you look for an existing property; that could work. Then you can renovate / improve it over the years.

Here I would probably go with the big providers after all. Work through the construction service description consistently and calculate the missing costs. NO upgrades, everything standard. Create the driveway with gravel, sow the lawn yourself. Plaster inside Q2, wallpaper with woodchip wallpaper. Floors laminate. It won't help, you have to go to the providers and talk to them. Don't talk about the budget, have them show you the individual options and give you the construction service description. 110sqm works with one child. But be aware that you will then only have one socket per corner of the room, no roller shutters, gas heating, etc. But you will have a house.

Hello, 600 € rent versus financing which is only slightly more expensive. You have to decide what you choose in the future. It is doable for you

So first of all: In our city and also in another one in the area, a whole settlement of terraced houses with 110sqm is being offered. They have 2 children's rooms with 13sqm, a bedroom of 15sqm and even an office with 9sqm. So you can definitely live with 110sqm!

Regarding the second point. With 350k you would probably find it hard to manage in our area, but I don't know how it is in RLP. Or as someone wrote before: no roller shutters, few sockets, no special features, no carport,... For us, a house was also a dream, more important than vacation or anything else, so I understand you!

But what I don't understand: In 10 years you'll be sitting there with 2 children in 110sqm without roller shutters, although the two of you will then have an income that would be enough for 140sqm with normal standards. Does that make sense? Why do you absolutely have to have the house alone?

We have 130 sqm of living space and will, if it is meant to be, also live in this house with two children. Exterior dimensions by the way rounded 7x11.

Just as a price orientation: We built in 2020/2021, solid wood without a basement, kfw40 with photovoltaic system. The pure house would have cost 330,000 without any upgrades. However, that was before the huge increase in material costs, which seems to have pushed it up quite a bit.