Question: How old was your AC unit? And how much did the compressor replacement cost? Because typically if your AC is out of warranty and your compressor goes bad, the best idea is to replace the entire unit (and you can usually write up the replacement as electrical damage and collect some insurance, because there is really no other reason for the compressor to go bad anyhow). I do AC work in my dad's business, and people around this area typically charge up to $1000 for a compressor replacement, and you can get a whole new AC unit installed for $1400 to $1500 with a brand new warranty, etc.
Thermostats themselves can not make fuses blow, they are not wired directly into your house wiring but rather into control modules on the furnace (or the furnace and AC depending on your setup). And they work on low voltage anyhow (~24 volts). (Unless you are talking about some little fuse in the thermostat circuit, but it does not seem that way.)
It sounds like you might have been having power supply problems if one day your compressor goes and then a couple days later a fuse randomly goes. It is possible that there could have been some mishaps with the wiring in the compressor replacement, but that is unlikely if it ever worked at all. Now though, with the furnace blowing fuses, it might seem to be some house wiring problems, but you would need more history to accurately analyze that.
As far as what Garfang said:
Can you call the people who did the original work right now? You should have a warranty on parts and labor.
You need to be careful about asking technicians to come back for warranty work on things that aren't actually warrantied and things that are warrantied but are not actually broken. First of all, your AC unit does not effect how your furnace runs (unless you are using a heat pump, but that's a different story). So, if you call up and
think your AC/compressor might be problematic, but you're really not sure, and you just want them to come out and
check things out, but it turns out that nothing is wrong with the compressor replacement, then you
will be charged a bill, because these guys don't do check ups for free. If you do want to do some preliminary checking though, I would just call them up and run your situation by them (but not indicating in any way that you are looking for a warrant cop-out, that'll turn sour quick) just to see what they think of your problem and what they think you might want to do with it.
Funny warranty story: We installed a furnace in a customer's house about 7 years ago, and the guy called up a little while back indicating that his furnace wasn't working and asking if we could come out and check it out. We said sure, set him up for a service call, and then the guy said "I'm not gonna be charged for this, right?", we stumbled for a second and then said yeh - the guy started getting irrate and we told him that if we warrantied the furnace out 7 years, we would have had to charge him twice as much for the installation - the guy just hung up. Eh, whatcha gonna do?
Edit: Okay, nm, looks like you have a heat pump, so most of what I said is just bull plop

- good luck with the capacitor. Make sure you try to discharge the old one before you take it out!
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