Is it halal to use the money you found on the street when nobody is around anymore? What if I send it to the police and then someone makes a fake claim?
No it is not. The money still belongs to the person who lost it and you do not have their consent to use it.
ياأيها الذين آمنوا لا تأكلوا أموالكم بينكم بالباطل إلا أن تكون تجارة عن تراض منكم
O you who have believed, do not consume one another's wealth unjustly but only [in lawful] business by mutual consent.
In short, you need to announce the found property for a year so that the owner can find it. After that you may give it as charity on behalf of the owner. If the owner is ever found you need to return it.
Regarding your concern of a fake claim, some details regarding the property are withheld and the claimant is asked about them to verify their ownership.
أن رجلا سأل رسول الله صلى الله عليه وسلم عن اللقطة قال عرفها سنة ثم اعرف وكاءها وعفاصها ثم استنفق بها فإن جاء ربها فأدها إليه
A man asked Allah's Messenger (ﷺ) about the Luqata. He said, "Make public announcement of it for one year, then remember the description of its container and the string it is tied with, utilize the money, and if its owner comes back after that, give it to him."
فإن جاء أحد يخبرك بعددها ووعائها ووكائها فأعطها إياه
If someone comes and informs you about the number (of articles) of the bag and the straps, then give that to him
That's basically stealing, which is strictly prophibited in Islam. Allah says: **“[As for] the thief, the male and the female, amputate their hands in recompense for what they earned [i.e. committed] as a deterrent [punishment] from Allah. And Allah is Exalted in Might and Wise.” [Quran 5:38] ** The reason the punishment is that, is because it scares people from stealing, meaning less people will steal.