2. If I do renew my visa, and stay a few months past when my original visa expires, would this hurt our application for a US spousal visa for my husband? As the expiration date would be at least 2 years from the time he'd hopefully be immigrating to the US...
So long as you are resident in the UK and your residency visa / permit is current and valid, you will be able to submit your I-130 via the US Embassy in London. The process is called Direct Consular Filing (DCF) and it is much, much faster than filing I-130s with the USCIS in the USA.
London no longer has a residency period requirement, so it doesn't matter how long you've been resident in the UK or how long your visa / permit has been valid. There's no guidance on how much validity you should have left at the time you submit your I-130 to London either. That said, the I-130 processing can take a few months so I'd personally advise you to have at least 3 months of validity left so you don't ever get caught out with an expired residency visa / permit while your I-130 is processing. After the I-130 is approved, your visa status in the UK no longer matters as far as the USCIS / Embassy is concerned.
London will want to see a copy of your current and valid residency visa / permit and additional proof of residency like utility bills, tax statements, rent or mortgage statements, etc. with your name on them.
If your visa expires in June, you have plenty of time left to submit your I-130 while resident on your current UK visa. Whether you should renew your UK visa really depends on WHEN you'd like to move to the USA. Do you have any ideas about that? I can help give you some ideas on timing your US visa applications and your UK renewal (if needed).