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When sending an email is there anything in that process that passes on the details (i.e. model and firmware version) of your router to the mail server or intermediary?
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When sending an email is there anything in that process that passes on the details (i.e. model and firmware version) of your router to the mail server or intermediary?
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Generally no. Sometimes information may be accidentally included, but when that happens, it's not the actual model and version – it's only the manufacturer, and only a rough hint at that.
At application layers, whether you're using a web browser or a dedicated mail app, it doesn't care about what router you are using and doesn't even have reliable means of finding out.
However, if you are using a dedicated mail app (such as Thunderbird) which builds the message on your computer before directly sending it off via SMTP, it's not uncommon for that message to contain either your LAN IP address, and/or your LAN hostname, and these can sometimes give hints as to what router is being used.
For example, if the message indicates that your LAN IP address is 192.168.88.x, it is very likely to be a Mikrotik router using its factory default DHCP settings. If the address is 192.168.43.x, then it is very likely to be a Wi-Fi tethered Android phone (meanwhile 192.168.42.x is USB-tethered Android). Finally, if the message indicates that your LAN hostname is "JimPC.dlinklan", then it's obviously a D-Link router.
Note that this does not apply to messages sent by malware running on the router itself – it only applies to messages sent by standard email apps running on connected PCs. It also isn't something that the router forces to add – it's just something the mail app adds on its own, for historical reasons.
At network layers, routers don't add anything to IP packets or TCP connections that is not necessary. There is no standard IP option nor TCP header option (metadata) which would indicate the router model, and I have not heard of any router which would try to add its own custom options either.
(Besides that, IP options don't actually work on the Internet – on quite a few occassions people wanted to use IP options for something useful and found that many networks just have a blanket ban on them, making such packets undeliverable. TCP options do work a bit better and are used e.g. for MP-TCP, but just adding them willy-nilly still would be unacceptable just for the technical problems alone.)
Routers also don't change the data carried in TCP connections. There are rare exceptions to this – namely FTP, which many home routers still tamper with, due to the public/private address problem – but the same issues do not apply to SMTP or HTTP, and even with FTP the changes do not actually reveal the router's model at all.
In addition, most SMTP connections you make are encrypted with TLS so the router couldn't change anything even if it wanted to. And if you're sending mail through a website using HTTPS, then the router doesn't even know which HTTPS connections are mail-related at all.
However, dedicated spam-filtering firewalls (often used on corporate networks) do understand and intercept SMTP as that is their job, and they frequently add extra headers to the message's contents (which you can see using "View Headers"). These headers often just indicate the spam score and such, but they also give clues as to the firewall's manufacturer.