Let's clear up some misinformation.
1. The English word "solder" actually derived from the
French soudure and the Latin-derived L was added back to the written word later. So as is usually the case when British and American pronunciation have diverged over time, it's actually the American usage that has retained the original form. (For example, the contrasting "a" sounds in can and can't in British English is often trotted out as an example of American "laziness" in English, when in fact, British English used the same "a" sound in can and can't until long after the American colonies were established. The contrasting "a" was a change that happened later in British English only.)
2. Do you pronounce the L in "walk", "talk", "half", etc? No? How about colonel? Lieutenant? Worcester? Cough? OK, so we've established that English doesn't have a 1:1 correspondence of spelling to pronunciation. (Cf.
ghoti) And what about the many English accents that turn L into a W sound in many positions? (Think
cold ->
cowd.)
3. American English doesn't use the verb "sod" you're thinking of (we do have the noun for the
pregrown grass to plant in the garden), so for us, it doesn't have the double-entendre it does for you.
To say sodder instead of solder would also mean you should also say sodid instead of solid. PS, not all Americans say sodder.
4. Ummm, no. As explained, since the word "solder" didn't come directly from
solidare, but from the (already L-less) French
soudure, there's no direct process that led to the loss of the L sound within English. Moreover, in your example, the word "solid" doesn't contain the "ld" combination at issue. If anything, it would be
solid ->
sod. But no way would it become
sodid.
5. Yep, some Americans pronounce the L in solder because they're pronouncing it phonetically. So you do hear it sometimes, but it's considered non-standard in American English.