Is your board a 5V or 3.3V output?
The Step Up is a 5V, the Arduino a 5V.
Either way, you need to feed it to the appropriate 5V or 3.3V pin on the Arduino, not the barrel plug (what you've labelled RAW, actually labeled Vin on the arduino). This will bypass the onboard linear regulator.
Good spot, thanks! Yes, I agree with you, no need for a further regulator.
Additionally, you should use a schottky (Low forward voltage) diode from your 5V or 3.3V to the VIN pin to avoid a possibly nasty inbalance.
What about an electrolytic capacitor? It doesn't provide the same protection of a diode, but should additionally provide some voltage buffer, something like a 100uF or greater. Is it a good idea?
The reason for the above is because the Vin (Barrel) input is designed to be 7-12V, and doesn't work well with less. And if you are stepping a 3.7V battery up that high, you are going to then burn the energy off anyways using the linear regulator. Since you are already using a regulator, you can skip using the linear regulator.
Yes, I agree, I'll use the 5V pin rather than Vin/RAW
Also, my experience with the LiPo Chargers are that they tend to drain the battery when not charging it. Do you have a way to disconnect it? Also, Sparkfun sells a all-in-one Charger/Step-up board for 3.7V batteries that they sell.
The plan is to have it as part of the design, now with cheap boards, on the final version directly on the board so that's not an option. I can try to measure the current drain from the battery during power off, but that might be tricky because of the probably very low currents involved... I might consider using a switch in case the current drained by the charger gets in the mA range, otherwise I believe it is an acceptable loss to maintain a low price.
About the Sparkfun one I started from there, but the cost is too high to get into the design as it is, so I looked at cheaper boards from China and I'll try to get somewhere from there. Looking at Sparkfun schematics it doesn't look they have any protection and the battery seems connected as it is in my design between the booster and the charger, may be the components they use are better and higher quality, but nothing fancy.