It's been a long time since I've futzed with Exodus, but if I'm not mistaken it has a mnemonic seed back up.  You don't have to export individual private keys, you can use the Exodus seed phrase backup to import the entire wallet.  This will include your master private key, which is used to generate all associated addresses.

Once you've written down the seed phrase, in Electrum click on File>New/Restore, enter a name then click Next>standard wallet>I already have a seed.  Enter the seed phrase in the field provide, then click the Options button.  Select Bip39 seed.

https://i.ibb.co/BCnnDKr/Capture.png

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Electrum and Exodus are only client software.  They just access the same information from the blockchain.  You can still make transactions in either, and they will both update once the information is sent to the blockchain.

Your ETH will continue to be accessible by Exodus.  You can import your ETH into other wallets the same way, with the same seed phrase...

Which brings me to something else:  Using a single seed phrase for multiple currencies isn't a very secure practice.  I recommend against it.  It provides more opportunities to have your seed compromised, which can jeopardies all your coins.

You many not have to transfer your bitcoin to Electrum.  If your current wallet is a web-wallet or a exchange wallet, then yes, you should transfer the money to your Electrum wallet.  But, if your currently using another desktop wallet you can just import your backup seed-phrase or the private keys into Electrum.  Doing so won't cost you anything in fees.

The "fee problem" occurs when you have to use more inputs to create a transaction, that'll make the transaction's size very high.
That will happen when you received too many small transactions to your wallet because each transaction's outputs that belongs to your wallet will be counted as an input; that includes the change (info. change).
Your wallet will need to use a number of those inputs depending on the amount you want to send.

In your described scenario, the first transaction will be consolidated to a single input as you sent all of your funds to Electrum.
The "small amounts" being a problem however depends if you're going to use Electrum to send funds (eg. to merchants), if each or two of those amounts are good enough to fill your typical transaction, then you don't have to consolidate them to a single input.
But if you're not planning to use them like that, then consolidating them is necessary when you've lots of them (View->Show Coins) when the average fee rate is low.

Use this to see the current average fee rate, the lower the bars, the lower the fee: https://jochen-hoenicke.de/queue/#0,2h
Info about consolidating inputs: Fees are low, use this opportunity to Consolidate your small inputs!

There shouldn't be much problem with the fee if it's just once a month, you'll only generate 12 unspent outputs per year;
but that depends on the amount of each monthly transactions and how you're planning to spend them.

If you're going to send all of the funds in one go, you should follow the reply above to consolidate your funds when the average fee is low.
But if your inputs (each 'deposit' or two) are large enough to fill your typical spends, then you don't have to consolidate.

If you have "unconfirmed transactions" in the history that are flagged as RBF (replaceable) and you're going to send another transaction, the one in the history will be combined to your new transaction.
That will enable you to save some fees by combining two transactions that's lower in size compared to two separate transactions.

Rbf is replace by fee. If a transaction has too low a few that the network can't confirm it quickly (or at all in some cases) or you need it confirming fast then you can use rbf on the tansaction(s) to speed them up.

It doesn't cost anything to enable rbf so you're better off leaving it enabled.

You can use different addresses to receive each time. The only problem with fees comes in the form of hsving to pay higher fees you you want to make a larger purchase than one of your inputs. It might be wise to consolidate your inputs (send all your funds to a single address in a single transaction in order to avoid this - when fees are low too).

There are many receive addresses to support airgapping, invoices and privacy. If you care about the latter you can also just use the "receive" screen.