This can easily be done and due to Bitcoin mechanics, you are going to be pretty safe. Electrum wallets cannot be desynced, maybe become inaccurate but the same seed will always give you access to the corresponding wallet and its bitcoins. Electrum inaccuracies in balance and transaction history rectify themselves over time. Just restore from seed on the other computers to set your wallet up on multiple computers.

Why does this work out?

Bitcoin does a neat thing with transactions and history. I won't go into all of it but the blockchain makes working between two computers easy. Your seed is the deterministic private keys for 10 addresses on electrum. This will always be true, on any machine, anywhere in the universe (unless a bug breaks it on a computer).

As long as you have the private keys, you have access to all funds sent to the corresponding public addresses of the private key. Even if the wallet is offline and detached, it can still access value. Imagine the electrum seed as a key to a coin safe, you can have as many duplicates of key as you want, as long as they are identical they will open the coin safe.

The balance of your addresses are purely derived from info available to anyone on the Bitcoin network, your balance is not dependent on your Electrum program, computer, or identity. If you sent me an address, I could tell you the balance. To get your balance, Electrum queries the many Electrum nodes to sync up. The balance can be wrong on multiple computers but give it a few seconds to correct itself.

1. Electrum is based on Python, an interpreted language and thus operating system-independent. As long as you can install Python and the required Python packages in the corresponding OS, Electrum should work.

2. When trying to figure out which addresses have been used, Electrum stops probing further addresses if several addresses (the gap limit) do not contain any BTC anymore (i.e. from transactions whose output has already been reused). So increasing the gap limit makes sure Electrum finds all generated addresses which contain any BTC. As far as I understand the code after a few minutes, when not using change addresses there is only one list of generated addresses, otherwise there are two lists.

3. Using an additional seeded/online wallet for small amounts is a trade off between security and comfort. E.g. if you have a local store which accepts BTC, using an seeded wallet on your smartphone is much more comfortable than running home and signing the transaction using your offline wallet. The same is true for a seeded wallet on your online PC, although "running home" may not be an issue there. Multiple wallets can be used by starting Electrum with the "-w walletfile" parameter (replace walletfile with one the corresponding wallet files).