Using a multimeter to check for power is frowned upon by the electrical industry due to the number of ways it can go badly wrong! Reading "phantom voltages" as you are reporting is really the least of your worries.
So if you are doing much in the way of DIY then getting a basic voltage tester is a very good idea. I prefer the sort that don't need batteries as it is one less thing to forget to change and find it has leaked and ruined it. Equally you
must test the tester every before using it! A professional would also buy a "proving unit" to provide various voltages on-demand to check the tester responds OK, but for DIY testing it on a known live outlet before checking something is not powered is a minimum step.
The issue of how good the earth rod is in a TT system is harder to check. A professional electrician would have equipment to measure that, most simply using a "multi function tester" (MFT) that can measure the earth fault impedance, but they cost around the £500-1000 region typically so not many DIYers are going to have access to one.
In the UK you can get better socket testers that do report high fault impedance, but they are not really designed for TT use as the majority of the UK is on TN-S or TN-C-S that has (or should have!) a very low and reliable earth impedance. For example:
A pocket sized checker for indicating loop impedance, socket polarity and RCD 30mA are all ok. Indication is by LED's and buzzer. An ideal unit for initial "checking out" of a 13A socket and will advise if the circuit is ok, needs attention, or needs urgent attention. An ideal product for...
www.toolstation.com
I don't know if a similar product is available in France to do that job with thresholds that are appropriate for TT installations, but maybe
@Mike Johnson will know as he has some experience of French electrical work.