Help with fist mini solar system

35ah is small, let's say you can use 20ah/day and recharge 20ah on a sunny day.

Let's ignore inverter inefficacy for a moment.

50w fan @ 12v is 4 amp so in 4-5 hrs you've used up all the 20ah from the battery.
75w TV @ 12v is 6+A or 3 hrs at best!

How much can a 100w panel make? I call 100w panel a 5A panel so it should make 15ah in winter, 20 spring/fall and 25 in the summer.
Pointing at the sun/moving it around and ya might do better.

A 400w inverter likely has poor eff, say 80%. so the above hrs of operation will be reduced.
Also note that there is a constant draw on a battery hooked to an inverter with no load so turn it OFF when not in use.
Inverters Low voltage cut off may also limit how low you can drain the battery
I'd stick to just one or two LEDs and limit their use!
One dreary/cloudy day and you are out of power!

irradiace multiplier for location, time of year and angle (use the results to multiply by 5A (100w panel) to determine your ah produced)
www.solarelectricityhandbook.com/solar-irradiance.html

On a sunny day, if the battery gets to 14.4v and stays there from enough sun, that might be a good time to use excess solar power.
A charge controller if it gets the batt up to 14.4v then limits power so as to not overcharge, you can use this extra power but if you use too mush the batt voltage will drop. Long term repeated undercharging will  limit the life of the battery, get it FULL at least every few days.

Even with adding a 100ah battery and just one panel it's still going to be minimal... You can do what you need in the day time if you have full sun but you will be very limited at night even if your careful...

I would go up to at least 400w in panels and at least 2 12v 200ah batteries... What Charge controller do you have??

You need to remember you really don't want to go below 55% of your battery charge and you can only use half of the power you can make in a day...

That's why I have the system I have today.. I also started with a small system.. Which was much larger than the one you have now. And smaller than the one that I suggested to you..

  I totally understand, but I was just being honest... You are going to be hard pressed to do what you want with such a small system no mater what you have been told...

Good luck with your situation...

I wanted to thank you for all the support you have given me. I'm trying my best to get a lead-acid deep cycle AGM battery, if I do not get 100ah at least a 70-90 battery would be great to increase the usage time. The panel is very well oriented, the controller kept me up until 6:15 p.m. Within two weeks I will be able to buy a 100w Monocrystalline expansion kit and thus have at least 10 amps of charge. Am I on the right way?

Renogy 100 Watt 12 Volt Monocrystalline Solar Expansion Kit features a 100 Watt Monocrystalline Solar Panel, a set of Zbrackets, and a pair of branch Connectors for $157.20 free shipping.

could you help me decipher if adding a panel and 100ah battery could help me use the led bulbs I need at the moment, a TV at least 2 hours during the day and a fan at night for at least 8 hours?

I also have an electronic timer that I can adjust to turn off the fan at a certain time. And there comes a voltmeter on the way that can help me easily see the voltage level that I have.

Can you estimate usage in amp-hrs? 75 watts @ 12v is 6.25 amps and each hr of use is 6.25 amp hrs plus the ineff of the inverter.
A 50w fan @ 12v is 4.2 amps, each hr of use is 4.2 amp hrs.

Can you add up the espected usage? You need to add in the inverter use when on and its eff with a load.

Did you go to the irradiance link and find how much sun ya get based on location, time of year, panel angle?

Is all this information a secret or can you share it?

If I could generate ~20 ah a sunny day and had an 100 ah battery and limit use to 20ah a day, on not so sunny days my battery could go a couple of days using 20ah a day, when the sun shined again the battery would NEVER get FULLY charged again if I use 20 and can make 20!
On sunny days I need to make more than I use so I can FULLY charge the battery. I could turn everything off for several days or I could make more than I use to get the battery FULL.