Is it possible to rebuild/make firmware for a router with a backdoor for attackers? I mean If attackers get access to a home router and they're make a new firmware with backdoor in case If I will change admin panel password?
Once a machine is compromised you can't even trust its "firmware update" feature anymore as you can't be sure it really does overwrite the current, malicious firmware. The only way to be sure would be to desolder the memory chips and rewrite them from a trusted computer or flash it via JTAG.
On the other hand, those firmware-reflashing attacks aren't common unless it's a targeted attack, so I would say you have nothing to worry about unless you have actual proof that your router was compromised.
Firmware-rewriting attacks on routers are theoretically possible, but they're not common. An attacker would need to tune the firmware to the exact model and hardware revision involved (for example, a WNDR 3700 v2 and a WNDR 3700 v3 are completely incompatible). Further, some routers require the firmware to be signed by the manufacturer; duplicating this signature can be extremely difficult.
It's much easier just to assume that the firmware bugs or weak passwords that let you in the first time will continue to work -- almost nobody pays attention to their router after setting it up.