LTC4040 Battery Backup Power Manager
According to the datasheet, the VIN to BAT path is a buck (step down) converter, and the BAT to VSYS path is a boost (step up) converter.  See page 13 for the description of the buck-mode battery charger.  See page 15 for the description of the boost-mode powering of the system via the battery.

The buck mode charger will not be able to charge the battery if the input voltage is less than the battery voltage.  So your first question:


  A typical application says 4,5v but can the IC charge the battery in natural charging mode (4,1V) using as input a voltage low as 3,5V?


The answer is no.  If the input voltage is 3.5V, it will not charge the battery to 4.1V.  I can't see any specifications that would provide a "minimum input voltage" for the buck-mode charger to charge to 4.1V, besides the "typical application circuit" calling out a 4.22V PFI threshold.  Although this suggests a 4.22V minimum voltage, targeting a 4.5V typical would be appropriate.  The specifications to look for would be "maximum duty cycle" for the buck converter, but this is not listed.  I would treat 4.22V as the minimum, or ask a factory representative for clarification based on your application.

For your second question:


  As it says, if input voltage is 3,5V then my battery charging up to 3,5V. Right?


The answer is no.  The voltage provided at the PFI input determines when the chip is in buck-mode (charging) or boost-mode (backing up VSYS with the battery); see the block diagram on page 12.

When PFI is > 1.19V, the LTC4040 passes VIN to VSYS, and charges the battery.
When PFI is &lt; 1.19V, the LTC4040 ignores VIN, and boosts VBAT to VSYS to power the system.

There is a tolerance to apply to PFI (PFI Input Threshold and PFI Input Hysteresis on page 4).